introduction to astronomy (coursera)

8/13/2019 Introduction to Astronomy (Coursera)

http://slidepdf.com/reader/full/introduction-to-astronomy-coursera 1/91

/91

Introduction to Astronomy

Notes from Olivier Henry ([email protected]), based on Coursera Introduction to

Astronomy course by Prof. Ronen Plesser

1. Coordinates on Earth: longitude/latitude.................................................................................62. Motion of the stars in the sky ..................................................................................................6

3. Seasons ....................................................................................................................................7

4. Horizontal coordinate system..................................................................................................8

5. Equatorial coordinate system ..................................................................................................9

6. Recap: Horizontal vs. Equatorial coordinate system.............................................................11

7. Localizing stars......................................................................................................................11

7.1. Using observer’s latitude, declination and altitude. ......................................................11

7.2. Exercise: maximum rise of a star based on latitude & declination ...............................12

7.3. Exercise: how high is the Sun at noon?.........................................................................12

8. Solar time vs. Sidereal time...................................................................................................13

8.1. Ignoring Precession and Nutation effects......................................................................138.2. Exercise: when will we see Vega at its highest point....................................................14

8.3. Precession effect............................................................................................................148.4. Nutation effect ...............................................................................................................15

8.5. Equation of Time...........................................................................................................15Part 1: Influence of Earth’s orbit around the Sun..................................................................16

Part 2: Influence of Earth’s inclination .................................................................................17

9. Motion of the Moon ..............................................................................................................18

9.1. The hidden side of the Moon.........................................................................................18

9.2. Eclipses..........................................................................................................................19

9.3. Partial, Total and Penumbral eclipses ...........................................................................2010. Planetary Motions..............................................................................................................21

10.1. Synodic period...........................................................................................................21

10.2. Three laws of Kepler .................................................................................................22

a) The orbit of every planet is an ellipse with the Sun at one of the two foci.......................22

b)A line joining a planet & the Sun sweeps out equal areas during equal intervals of time.22

c) The square of the orbital period of a planet is directly proportional to the cube of the

semi-major axis of its orbit....................................................................................................23

Example: period of the ISS ...................................................................................................23

10.3. Centripetal force ........................................................................................................23

10.4. Newton's law of universal gravitation .......................................................................24

10.5. Law of conservation of momentum...........................................................................2410.6. Potential energy.........................................................................................................2510.7. Law of conservation of energy..................................................................................25

Exercise: finding the thermal velocity...................................................................................25Exercise: finding the escape velocity ....................................................................................25

10.8. Tidal forces................................................................................................................25

11. Waves ................................................................................................................................27

11.1. General definitions ....................................................................................................27

11.2. Doppler effect............................................................................................................27

11.3. Light ..........................................................................................................................27

11.4. Heat transfer and radiation ........................................................................................27

Heat transfer ..........................................................................................................................27Radiation ...............................................................................................................................28

Luminosity is in W (=J/s) ......................................................................................................28



/91

Exercise: Flow produced by Sun at its surface......................................................................28

Exercise: Temperature of the Sun .........................................................................................29

Exercise: color of the Sun .....................................................................................................29

Exercise: finding the speed of an helium particle, knowing temperature .............................29

12. Electromagnetic force........................................................................................................29

13. Solar system ......................................................................................................................30

13.1. General ......................................................................................................................3013.2. The Sun......................................................................................................................3113.3. Interplanetary medium ..............................................................................................31

13.4. Age of the Solar system.............................................................................................3213.5. Nuclear/radioactivity decay.......................................................................................33

13.6. Radiometric dating ....................................................................................................3313.7. Radiometric dating (Uranium-lead) ..........................................................................33

13.8. The creation of the solar system................................................................................3413.9. Kelvin–Helmholtz contraction: gravity contraction! heat ......................................35

13.10. Terrestrial formation..................................................................................................35

13.11. Beyond the Snow line................................................................................................36

13.12. Orbital resonance.......................................................................................................36Kirkwood gap ........................................................................................................................36

The Nice model .....................................................................................................................36

13.13. Timeline of the creation of the Solar system.............................................................37

14. The Earth ...........................................................................................................................38

14.1. General ......................................................................................................................38

14.2. Internal Heat ..............................................................................................................38

14.3. Finding temperature of Earth ....................................................................................38

Ignoring Earth’s atmosphere .................................................................................................38

Greenhouse model.................................................................................................................39

14.4. The Atmosphere ........................................................................................................39

14.5. Earth magnetism........................................................................................................40

15. The Moon ..........................................................................................................................4116. Planets detection methods .................................................................................................43

16.1. Astrometry.................................................................................................................4316.2. Radical velocity.........................................................................................................43

16.3. Transit method...........................................................................................................4316.4. What have we found?................................................................................................44

17. Analyzing Stars .................................................................................................................45

17.1. Laws of conservations of charge and of electrons ....................................................45

17.2. Chemical reactions to create heat ..............................................................................45

17.3. Nuclear Fission to create heat ...................................................................................4517.4. Nuclear fusion to create heat: PP chain.....................................................................46

Exercise: quantity of He produced since the Sun’s birth ......................................................46

17.5. Solar Structure...........................................................................................................47

The core.................................................................................................................................47

Inner Mantle ..........................................................................................................................47

Outer Mantle..........................................................................................................................47

Chromosphere .......................................................................................................................47

Corona ...................................................................................................................................47

17.6. Solar weather.............................................................................................................4817.7. Parallax, or Finding distance from the Star ...............................................................49

17.8. Astrometry, or finding the speed of the Stars............................................................5017.9. Stellar statistics ..........................................................................................................51

17.10. Binary stars................................................................................................................51



/91

17.11. Eclipsing binary stars ................................................................................................51

17.12. Recap: Analyzing Stars .............................................................................................52

By visual observation ............................................................................................................52

Using Doppler effect .............................................................................................................52

Particular case of Eclipsing binaries .....................................................................................53

17.13. Mass-luminosity relation...........................................................................................53

17.14. Main-sequence stars ..................................................................................................53This is our Sun.......................................................................................................................53CNO cycle for MS stars > 1.3 R sun........................................................................................53

Radiation and Convection effects..........................................................................................54Expansion by contraction ......................................................................................................54

18. Star evolution ....................................................................................................................5518.1. Star creation process..................................................................................................55

18.2. T-Tauri stars ..............................................................................................................5518.3. Main Sequence ..........................................................................................................56

18.4. From MS to Subgiant stars ........................................................................................56

18.1. From Subgiant to Red giant branch stars ..................................................................56

18.2. From Red giant branch to Helium core flash ............................................................5718.3. From Helium core flash to Horizontal branch...........................................................57

18.4. From Horizontal branch to Asymptotic Giant branch...............................................58

18.5. From Asymptotic Giant branch (AGB) to Thermal Pulse AGB...............................58

18.6. From Thermal Pulse AGB to White Dwarf...............................................................58

18.7. White Dwarf Nova ....................................................................................................59

18.8. Supernova (type Ia) ...................................................................................................59

18.9. Instability branch: Variable stars as Standard candles ..............................................60

18.10. Blue stragglers ...........................................................................................................61

18.11. From MS to Red Supergiant......................................................................................61

18.12. From Red Supergiant to Helium flash to Blue Supergiant........................................61

18.13. From Blue Supergiant to Massive star AGB.............................................................62

18.14. From MS to Wolf-Rayet stars ...................................................................................6318.15. From MS to LBV stars ..............................................................................................63

18.16. From Core collapse to Supernova type-Ib/Ic/II.........................................................6318.17. From Supernova type-Ib/Ic/II to Neutron (pulsar) star .............................................64

18.18. Recap: Stars on HR diagram .....................................................................................6519. Relativity ...........................................................................................................................66

19.1. Principle of Relativity ...............................................................................................66

19.2. Spacetime ..................................................................................................................66

19.3. Lorentz transformations ............................................................................................66

19.4. Relativistic Spacetime ...............................................................................................6719.5. Length contraction.....................................................................................................67

19.6. Time dilation .............................................................................................................67

19.7. Doppler effect due to high speed...............................................................................67

19.8. Velocity addition .......................................................................................................67

19.9. Lorentz metric ...........................................................................................................67

19.10. The Invariant Interval ................................................................................................68

Time-like interval ..................................................................................................................68

Light-like interval..................................................................................................................69

Space-like interval .................................................................................................................6919.11. Conservation laws .....................................................................................................69

19.12. Lorentz transformations applied to Energy and Momentum.....................................6919.13. Principle of Equivalence ...........................................................................................69

19.14. Gravitational redshift.................................................................................................70



/91

19.15. Relativistic Potential energy......................................................................................71

19.16. Gravitational lensing .................................................................................................71

19.17. Gravity is geometry ...................................................................................................72

19.18. Gravitational waves...................................................................................................72

20. Black holes ........................................................................................................................73

20.1. Horizon......................................................................................................................73

20.2. Singularity .................................................................................................................7320.3. Emission of X-rays....................................................................................................7320.4. No Hair ......................................................................................................................74

20.5. Cosmic censorship conjecture ...................................................................................7420.6. Hawking radiation .....................................................................................................74

20.7. Wormholes ................................................................................................................7420.8. Example: compute wavelength of X-ray emission of the accretion disk surrounding

black hole ..................................................................................................................................7521. Galaxies .............................................................................................................................76

21.1. The Milky way ..........................................................................................................76

21.2. Tracking matter .........................................................................................................76

21.3. The Milky way disk structure....................................................................................7621.4. The Milky Buldge and Core......................................................................................77

21.5. The Milky Halo .........................................................................................................77

21.6. Weighting the Milky way..........................................................................................77

21.7. Dark matter................................................................................................................78

21.8. Spiral galaxies ...........................................................................................................78

21.9. Galactic evolution......................................................................................................78

21.10. Measuring distance to galaxies: Redshift ..................................................................79

21.11. Cosmic expansion......................................................................................................79

21.12. Recap on formulas.....................................................................................................80

21.13. Galaxy clusters ..........................................................................................................81

22. Cosmology.........................................................................................................................82

22.1. The cosmological principle .......................................................................................8222.2. Robertson-Walker model ..........................................................................................82

22.3. Angular size distance (k=0).......................................................................................8222.4. Luminosity distance (k=0).........................................................................................83

22.5. Correcting the temperature for redshift .....................................................................8322.6. Correcting the galaxy speeds for redshift..................................................................84

22.7. Einstein field equations .............................................................................................84

22.8. Isotropic Homogenous Matter...................................................................................85

22.9. Friedmann equations .................................................................................................85

22.10. Cosmological parameters ..........................................................................................8622.11. The Early universe: radiation era ..............................................................................86

22.12. Matter-dominance era................................................................................................86

22.13. Dark-energy-dominance era ......................................................................................86

22.14. The Particle horizon ..................................................................................................87

22.15. The event horizon......................................................................................................87

22.16. Cosmic microwave background ................................................................................87

CMB ......................................................................................................................................87

Angular Power Spectrum of the CMB ..................................................................................88

22.17. Big Bang Nucleosynthesis.........................................................................................8822.18. LCDM Cosmology....................................................................................................89

22.19. Inflation .....................................................................................................................8922.20. Exercise: compute the distance when the light was emitted, and the distance now,

from a galaxy.............................................................................................................................90



/91

Highest point, fromobserver’s view

1. Coordinates on Earth: longitude/latitude

Longitude is going east.

2. Motion of the stars in the skyBecause of the Earth’s rotation, starts are moving with an ! angle which depends on thelatitude of the observer (Stars – except the Sun - are so far from Earth that they seemfixed): ! = 90° - latitude. Hence:

• For an observer on the pole,starts go on a path parallel to thecelestial equator

• Stars rise and descend• The Sun moves along the

Celestial sphere from West toEast (RA increases), becauseThe Earth rotates in the samedirection as it orbits around theSun. Within a year we see allstars.

Spring/Summer on Northern

Hemisphere

Spring/Summer on Southern

Hemisphere



/91

Special case when "=90°:

• We can use the starts to find our latitude: the angle above our head to which wesee the Polaris star corresponds to our latitude on Earth.Polaris (or ! UMi, or ! Ursae Minoris, or Alpha Ursae Minoris) is a North Star(also called Pole Star) in the constellation Ursa minor (“petite ourse” in French),very close to the celestial pole.

• Also, stars located at the observer’s Zenith have declination = latitude.

ZenithThe zenith is an imaginary point directly "above" a particular location, on the imaginarycelestial sphere. The zenith angle is the angle between a direction of interest (e.g., astar) and the local zenith.

3. SeasonsSeasons are due to the inclination of Earth (24°) vs. the Sun.

• At solstice, the day is the longest or shortest.“Solstice” comes from Latin “sol” (sun) and “stilium” (stoppage) because from oneday to the next, the Sun seems to stay at the same place vs. Earth.

• At Equinox, day and night are about the same length: 12 hours, except on poles.On poles, days are 6 months, nights are 6 months, changing at solstice. In fact,this would be true if the Sun was just a point. But since the Sun is seen fromEarth as a sphere, in practice the day is longer by a few minutes (depending onthe latitude).

At equinoxes, Sun rises vertically. Vernal equinox is ~ March 21, Autumnal equinox is ~September 21. June solstice is ~ June 21, December solstice is ~ December 21.



/91

Earth is inclined 23.5°, which1. Creates seasons2. Creates the impression from

Earth that the Sun is orbitingin an elliptic path, called theEcliptic.

The Earth rotates in the same direction as it orbits around the Sun.

4. Horizontal coordinate systemThis system is using:

1. azimuth (angle from magnetic North)2. altitude (height of the star in the sky)

The star’s altitude and azimuth change through the night and depend on the observer’sposition.



/91

5. Equatorial coordinate systemThis system is using:

1. Right ascension is the longitude (going east). But not starting from Greenwich,rather from the Vernal Equinox. Degrees are converted in hours minutes seconds(24h = 360°)

2. Declination is the latitude of the star if projected on Earth3. RA always remain the same for a Star, except for close Stars like the Sun. RA

varies between -23.5° and +23.5° for the Sun, through the year.

This system does not depend on the observer’s position or time.

• 1 degree = 1hour of arc• 1 hour of arc = 60 minutes of arc = 3600 seconds of arc• Thumb is ~ 1 degree at arm’s length• Hand is ~ 20 degrees at arm’s length

For two stars one hour of right ascension apart, you will see one star cross yourmeridian one hour of time before the other.

The start of the RA (Vernal equinox) can be easily visualized using the Pisces

(“poisson”) constellation:

Hour °

-23.5° RA

+23.5° RA

0° RA

Hour °

20°



/91

Remember that from Earth, in this coordinate Stars seem fixed, because celestialsphere very big.



/91

6. Recap: Horizontal vs. Equatorial coordinate systemHorizontal coordinate system Equatorial coordinate system

(azimuth, altitude) (RA, Declination)

7. Localizing stars

7.1. Using observer’s latitude, declination and altitude.

Latitude = declination + ZA or

Declination-ZA depending on position of star vs. observer

! ZA = |latitude – declination| and Altitude = 90°-ZA

ZA = Zenith Angle(angle between star and observer’s

zenith), seen from observer



/91

8. Solar time vs. Sidereal time

8.1. Ignoring Precession and Nutation effects

By definition, 24h is the time needed for the Earth to see the Sun come back at thesame place. This is called the solar time. But that ignores the fact that during this time,

the Earth has been orbiting around the Sun. Hence the Earth rotates in less than 24h.This is called the sidereal time.

1. Earth orbits around the Sun in 365.25 days! !24 = 360/365.25 ° per day, or per24h.

This is the degrees by which the Earth orbits the Sun in 24h

2. Earth rotates 360° + !24 in 24h360° in 360/(360 + !24) x 24h

! Earth rotates 360° in Tsideral = 360/(360 + !24) x 24 = 23h 56’

in Tsideral = 23h 56’ Time for Earth to do 360° rotationTsolar = 24h Time for Earth to view Sun at the same place

This is equivalent to Tsolar - Tsideral = 1/366.25 days = 24x60/366.25 " 4 min (3.83 min) since in one year, the Earth rotates 365 times relative to the Sun, but 366 times relativeto the stars.

• That is why Stars (including therefore the Sun) rise 4 minutes earlier every day.• Hence in one year, the Earth has rotated 365.25x24/23.93356 = 366.2 times. Hence

stars shift slowly with every year. This effects adds to the precession effect.• Solar time is called local time • On Sep 21, solar time and sidereal time are the same• ST = LT +/- 4minutes

Sidereal time= 0 at vernal equinox (June 21st

). Any celestial body is crossing the localmeridian at its right ascension.



/91

Earth rotates 366.25 times @ 23h56’, or 365.25 times @ 24h. Therefore, sidereal timefor Earth is 365.25 days (of 24h).

Local Sidereal Time

Sidereal Time (“Greenwich Sidereal Time”) is the time such as a day is 23h56 min, and starts

(ST=0) at Vernal equinox.

Local Sidereal Time is the Sidereal time, but depending on the observer’s location.

Local Sidereal Time = Greenwich Sidereal Time + observer’s longitude in hours (360° = 24h)

Local Sidereal Time vs. local (Solar) time

• On September 21: local time = Local Sidereal Time. Then # = 4 minutes per day

• On December 21: local time = Local Sidereal Time - 6h

• On March 21: local time = Local Sidereal Time - 12h

• On June 21: local time = Local Sidereal Time - 18h

Each star is at its highest point (= on our meridian) when Local Sidereal Time = RA.

8.2. Exercise: when will we see Vega at its highest point

When is Vega as high as possible (from our point of view) at midnight?

Vega’s RA is 18h36’.

Vega is on our meridian at Local Sideral Time = 18h36’. (ignoring latitude and time zone

effects)

1. On June 21

st

, LST = LT + 18h therefore at LT = 0 on June 21

st

, LST= 18h and Vega is nearly on our meridian.

Hence Vega is at its highest point at midnight LT, on June 21st + 9 days (36’ = 9 days x 4’) =

July 1st.

2. We could also do: 18h36 = 18+36/60 = 18.6 h. Since 24h $ 365.25 days, 18.6h $ 283 days.September 21 + 283 days = July 1st.

3. We could also use the daily gap between Local Sidereal and Local Time (3.83 minutes). But

we need to remember that this value is based on 366.25 days, not 365.25 days, so we need to

withdraw 1 day: 18h36 = 1116’ = 284 x 3.83 minutes ! September 21 + 283 – 1 days = July 1st.

8.3. Precession effect

When the Earth is orbiting around the Sun, androtating around itself like a spinning top, the gravityfrom the Sun attracts the weight excess locatedaround the Earth equator closer to the Ecliptic. Thiscause the Earth’s axis to slowly move in a coneshape, as a spinning top would do (rotation of theaxis in the opposite direction of the spinning toprotation).

Rotation of the axis rotates 360° in 26000 years,hence 1.4° every 100 years.

on Dec 21 0:00 ST, it’s not

yet Dec 21 LT because

Tsideral = 23h 56’



/91

Because of this effect, Vernal equinox advances slightly every year, hence the name“precession of the equinoxes”.

8.4. Nutation effect

The Moon also creates such effect, called Nutation.

Period is 18.6 years.Oscillation is 17,2" = 17.2/3600° = 0.005°

Precession + Nutation effect

8.5. Equation of Time

• The Earth’s orbit around the Sun is not a circle, for several reasons. Because ofthis, the Sun is not moving at a regular speed around the Ecliptic. The equation

of Time indicates the difference between the time viewed from a sundial (“real”)and the official time (or “apparent” because based on the assumption that theSun is moving at regular speed on the Ecliptic).

• The sundial indicates the real time, whereas our clocks indicate the apparenttime (= average)

Real time = Apparent time - !T(d) where d is the day (d=1 for Jan 1st).

The real equation of Time in minute is:

#T(d) = 4 x [C(d) + R(d)] where C and R are expressedin degree, such as:

C (d) = 1.918° sin(d) + 0.02° sin(2d) + 0.0003° sin(3d)R(d) = -2.468° sin(2d) + 0.053 sin(4d) – 0.0014° sin(6d)

This can actually be approximated as:!T(d) = !Tc(d) + !Tr(d)

#Tc (d) = 7.678 sin (B+1.374)

#Tr (d) = -9.87 sin (2B)



/91

Where B(d) = 2" (d-81)/365 (d=81 is the Spring Equinox)

Part 1: Influence of Earth’s orbit around the Sun• Earth’s orbit is not really circle, it’s a little elliptic• Earth’s speed is not constant on the elliptic path

• This effect account for up to 9 minutes difference in the real and apparent time.

(Earth’s orbit is greatly exaggerated on the drawing. Speed varies from 30287 km/s to29291 km/s)

Perihelion happens around Jan 4, Aphelion happens around July 4.

!Tc (d) = 7.678 sin (B+1.374)

Earth is fastest Earth is slowest

Part 1 Part 2 Equation of time = sum



/91

Part 2: Influence of Earth’s inclination• Earth is inclined, therefore the Sun’s projection on the Ecliptic is not linear• This creates a difference between the real time and the apparent Sun

!Tr (d) = -9.87 sin (2B)



/91

9. Motion of the Moon

9.1. The hidden side of the Moon

The Moon rotates in 27.323 days. This is its sidereal period, also called its siderealmonth.

Which means RA of the moon increases by 48’ per day (vs. 4’ for the Sun).

Because the Earth is moving around the Sun, the Moon’s full rotation relative tothe Sun is actually longer. This is called the synodic month = 29.53 days.The synodic period drives the full moon cycle.

The sidereal period is also the time that the Moon takes to rotate around the Earth ! from Earth, we always see the same side of the Moon. The other side will alwaysremain hidden from Earth. We say that the Moon’s rotation period and orbital period are

the same.

This is due to the tidal forces (“forces de marée”) applied for the Earth to the Moon, asindicated in the figure below (where the Moon is called the satellite).Let’s assume that the Moon is rotating fasterthan it is orbiting around the Earth. The Earth’stidal forces create 2 small deformations (oneon each side of the Moon), as indicated in (1).When the Moon is rotating, those deformationsare rotating as well, and are now “in advance”of the Earth. The Earth’s tidal forces apply to

those deformations, which tends to slow therotation of the Moon by forcing them to go“backward” as indicated in (2). With time, thesatellite will be rotating at the same speed as itis orbiting around the Earth.The same reasoning is valid also if the satellite was rotating slower than it is orbitingaround the Earth.

The tidal forces: the force on the left isstronger than on the right, making it seem as if

the planet was exposed to two opposedforces.

If the satellite was rotating faster than its orbiting speed, the tidal forces will alsomake it come closer to the planet. If the satellite was rotating slower than its orbitingspeed, the tidal forces will make it go farther from the planet. In the case of the Earth-Moon system, the Moon goes farther from Earth by about 3.8cm per year.



/91

9.2. Eclipses

Eclipses happen when the Sun, the Earth and the Moon are more or less aligned. Wespeak about solar eclipse when the Moon masks (partly) the Sun, and about lunareclipse when the Moon passes directly behind the Earth into its umbra (shadow).Eclipses do not happen every 27 days, though. This is because the plan of the Moon

rotation around the Earth is inclined vs. the plan of the Earth rotation around the Sun:The Moon’s orbit is tilted by 5° with respect to the ecliptic (it varies between 5° and5°18’ in 173 days).Like the Sun, the Moon is higher in the Summer. And both have almost the sameangular size.

Represented as a drawing:

Hence, a lunar eclipse happens roughly 2 times a year. Since the Moon orbits in 27days, it has done half a resolution around Earth in about 2 weeks. Hence, solar eclipse(most of the time, partial) and lunar eclipse happen roughly 2 weeks at interval. In total,there are therefore about 4 eclipses (lunar and solar) per year. This is why when perfectalignment, for a particular region on Earth (250km shadow), we can have total eclipses.

Practically, solar eclipse happens 2 weeks before the lunar eclipse.

In fact, using the notion of saros (18.6 years interval, where the Earth, Sun andMoon have exact same position), we can compute that there are about 4.6 eclipsesper year. This is because lunar and Earth orbits are not multiple of each others.

Lunar eclipse

Solar eclipse

Nothing

Nothing Full moon



/91

10. Planetary Motions

10.1. Synodic period

The synodic period is the temporal interval that it takes for an object to reappear at the

same point in relation to two or more other objects, e.g., when the Moon relative to theSun as observed from Earth returns to the same illumination phase.

The Synodic period of two planets can be easily found bysolving:

where S is Synodic period (i.e. when both align again), P1the planet’s period of the faster planet, P2 the planets’period of the slower planet,

Remember that where V is the planet’s speed.



/91

c) The square of the orbital period of a planet is directly proportional to thecube of the semi-major axis of its orbit.

P2 = K a3 where K is constant.

Or more precisely: whereM1+M2 is the total mass of the system

Where P is the orbital period of the planet orbiting and M2 its mass, M1 the other planet (Sun)’s

mass, and a the semi-major axis of the orbit. G is the gravitational constant. This relation is due

to the centripetal forces.

Because M1 >> M2, K is constant for every planet of the Solar system.

From this, we can get the planet’s speed for eclipses closed to circles (a%R) using P=2&R/v:

P2 = a

3 if units expressed in AU and years, since for Earth, using those units we find out

K=1. a = R is circle instead of ellipse. VALID FOR SUN ONLY

For other Stars, express Motherstar = '. Msun, then P2 = K. a3 where K=1/ ' using

AU and years.

Example: period of the ISSISS orbits at an altitude h = 370 km, Earth has radius of 6471 km and mass of 5.9272 x 10

24.

Hence P2

= 2 &2 R

3 / GM ! P = 5510s = 91.8 m.

10.3. Centripetal force

In simple terms, centripetal force is a force which keeps a

body moving with a uniform speed along a circular path and is

directed along the radius towards the centre. For a satellite inorbit around a planet, the centripetal force is supplied by

gravity.

Centripetal force

F= m.a where m is the mass in kg and a the centripetal

acceleration.

(N)



/91

Attraction force

On Earth, F= m.g attraction force, where g= 9.82 ms-2 constant for Earth.

Notice that the smaller r, the highest the F and the v.

10.4. Newton's law of universal gravitation

Every point mass attracts every single other point mass by a force pointing along the line

intersecting both points. The force is proportional to the product of the two masses and inversely

proportional to the square of the distance between them:

where:

• F is the force between the masses

• G is the gravitational constant = 6.67 x 10

-11

N.m2/kg2

• m1 is the first mass

• m2 is the second mass, and

• d is the distance between the centers of the masses.

It can be shown that the Earth is completely equivalent to a point of same mass, concentrated inthe middle. Using the above formula, we can compute the effect of gravity on someone at the

surface of Earth of 59kg: 579N.

10.5. Law of conservation of momentum

We define momentum as: p = m.v

The momentum represents how easy/hard it is to modify an object’s course.

In a closed system (one that does not exchange any matter with the outside and is not acted

on by outside forces) the total momentum is constant. pA + pB = constant This is particularly interested in the case of collision, when an object stops

and the other starts moving.

Notice that since F=m.a and p=m.v, F is the rate of change of p.

The angular momentum is also conserved : L = m.v.R if moving in a circle.

The conservation of angular momentum explains the angular acceleration of an ice skater as she

brings her arms and legs close to the vertical axis of rotation: R (, so v).

If a planet is found to rotate slower than expected, then astronomers suspect that the planet is

accompanied by a satellite, because the total angular momentum is shared between the planet

and its satellite in order to be conserved.



/91

10.6. Potential energy

Potential energy is the energy of an object or a system due to the position of the body or the

arrangement of the particles of the system.

For an object subject to gravity: where ME and Mo are the masses of Earth

and of the object.

Note that if object is located on earth, the Potential energy (in this case, energy is I release the

object) becomes:

So to sum-up:

where constant for Earth

10.7. Law of conservation of energy

Energy is constant, if no other forces than gravity is applied to an object.

Exercise: finding the thermal velocityThis is the speed of atoms excited by a certain temperature: Ec = % mv2 = 3/2 kB T

Exercise: finding the escape velocityConservation of energy ! Ec = * mv2 = -G. ME m/(R E+h) ! v

Comparing Thermal and Escape velocity, we can conclude whether those atoms remain in the

atmosphere, or are expulsed by temperature.

10.8. Tidal forces

Using gives: therefore is the

acceleration on Earth due to the Sun, or Tidal acceleration, which varies depending on where we

are on Earth:

g



/91

For there we get aT = a+ - a- !

This can be also expressed as:

= 5.14 x 10-8

g

This is a rather small value. What about the tidal forces due to the Moon?

Applying the formula to the moon, we get aTmoon = 2.2 aTsun. Tides repeat every 24h44 min.

There’s a 12 min lag.

This effect is even increased during full moon, when Sun and Moon are aligned with Earth. Atquarter moon, the tide is the smallest.



/91

11. Waves

11.1. General definitions

• f is the frequency, in Hz

• + is the wavelength, in m

• Energy flux transported by wave is proportional to Amplitude2 , in (J/s)/m2, or W

+ .f = c where c is the speed at which the wavelength is traveling.

11.2. Doppler effect

Where vrec and vem are the speeds of the receptor or emitter.

Or, if receptor is not moving:+ = + 0 (1-v/c) where c is the wavelength’s speed, and v the emitter’s speed.

Notice that the frequency does not change with time: either it’s less than f (if emitter gets

away from receptor), or it more than + (if emitter gets closer to receptor).

The sound effect we see when a car passes by comes from the amplitude difference with time.

The frequency is higher, but does not change with time.

11.3. Light

Light carries energy at a speed of c = 2.998 x 10^8 m/s

Color is the frequency of light, of the order of 10^12 Hz for visible colors.

Our eyes are only sensitive to the intensity of Red Green Blue (RGB) colors.

Because each atom absorbs particular frequencies, by looking at the spectrum of light emitted by

stars, we can find which atoms are present.

11.4. Heat transfer and radiation

Heat transferAn object hotter than environment will lose energy until temperatures equilibrate. It can happen

by:

- conduction, i.e. through continuous contact- convection, i.e. through physical motion

- radiation, i.e. hot objects glow losing energy to light. If energy is radiated at a rate L in J/s, atdistance R



/91

RadiationRadiation flow is distributed uniformly on surface on a sphere F=L/(4 " R

2)

• A hot object radiates

• The hotter the object, the smaller wavelength (gets blue): # max . T= b where b is

constant b=2.9 x 10-3

m.K T in Kelvins, b in meter x Kelvins

Hotter objects radiate more: F = $.T4 where F is the flux (i.e. power/m2) radiated, and

, = 5.67x10-8 W/m-2K -4 Stefan-Boltzmann constant

Sunlight heat on earth is the solar constant b0 = 1361 W/m2

From this we can compute the luminosity of sunlight (L is fixed) L = 4 " d2 b0 = 3.83 x 10

26 W

Flow is in W/m-2

Luminosity is in W (=J/s)

Energy captured on Earth (in W/m2) = L/(4&D2)

Energy radiated by Sun (in W/m2) = F.(4&R sun2)

Exercise: Flow produced by Sun at its surfaceFlow from Sun at surface is: F = L/(4& R 2) and Luminosity L=4 & d2 b0

Therefore F= (d/R)

2

. b0 = 6.29 x 10

7

W/m

2

Wien’s law



/91

Exercise: Temperature of the SunTemperature of the part of the Sun that we see, at its surface: F = $.T

4 ! T=5770 K = 5500 °C

Exercise: color of the SunUsing Wien’s law: + max = b/T = 0.0029/5770 = 503 nm. This corresponds to the green color.

Why not yellow?

The light emitted by the Sun if refracted by the Earth’s

atmosphere. First, the blue is refracted (which is why the Earthlooks blue from space). The Sun therefore looks yellow. Then, at

sunset, the distance that lights covers in the atmosphereincreases, and more light gets reflected – the sun looks red.

The highest frequencies get reflected first.

0K = -273.5°C

Exercise: finding the speed of an helium particle, knowing temperatureUse Ec = * mv2 = 3/2 k B T

12. Electromagnetic force

(in Coulomb) to be compared with gravity force.

• Force can be attractive or repulsive.

• Opposite charges attract to most objects are neutral.

• Charge is conserved

• A charge creates and is affected by electric field

• A changing magnetic field creates electric field (Faraday 1831)

• A changing electric field creates magnetic field (Maxwell 1861)

• This leads to propagating waves with velocity c speed of light ! light is an

electromagnetic wave!

Many waves’ frequencies are blocked by atmosphere, so we need to observe from space.



/91

13. Solar system

13.1. General

Our solar system is composed of:• the Sun (99.86% of total mass of the solar system)• 8 planets: Mercury, Venus, Earth, Mars, Jupiter and Saturn (90% of the

remaining solar system mass after the Sun), Uranus and Neptune. Distancesfrom the Sun ranges from 0.39 – 30 AU

• Their 175 natural satellites (or “moons”), most of them orbiting Jupiter and Saturn• 5 Dwarf planets: Pluto, Ceres, Eris, Makemake, Haumea• Billions of small bodies• All orbits are in the same plane, the Sun’s axis

All planets, as well as most other objects (except the Halley comet), orbit around theSun in the same direction as the Sun’s rotation: counter-clockwise for an observerlocated on the North pole.

All objects orbiting around the Sun do so in an elliptic path, from which one focus is the

Sun. Planets’ orbit is nearly circular, while the smallest the other object, the more ellipticthe path.



/91

Kepler's laws of planetary motion are three scientific laws describing orbital motion,each giving a description of the motion of planets around the Sun.

Most of the largest natural satellites are in synchronous rotation, with one facepermanently turned toward their parent. All the 4 giants have rings. A planetary ring isassumed to be quite instable, and disappear after a few thousand or millions years.

Hence today’s planetary rings are quite recent. One object from the ring is eitherattracted back to the ring, and therefore stays in the ring, or attracted by the planet (andtherefore disappears within the planet). Hence rings have distinctive edges.

• 70.5% is Hydrogen, 27.5% is Helium, 2% is Metal

13.2. The Sun

The Sun is a yellow dwarf, as 20-40 other billions yellow dwarf in the Milky way (for atotal of 200-400 billions starts).Each second, the Sun merges 564MT of Hydrogen and produces 560MT of Helium.The difference, for a weight of 4MT, produces energy and is radiated as light and solarwind. Every 150M years, the sun looses the equivalent of 1 mass of Earth.

The Sun is in its mid-life. In 5 bn years, it will become bigger, more bright, colder, morered: a red giant. It will then be several thousand times more bright than today.

The Sun is a star Population I: it is born from supernovae’s explosions, which createdheavier metal. It is widely assumed that the presence of heavier metal in the Sun isrequired to form planets, grouping metals together.

13.3. Interplanetary medium

In addition to light, the Sun alsoradiates a continued flow ofcharged particles (a “plasma”)called “solar wind”. This flowsextends at a speed of 1.5M kmper hour, creating an atmospherecalled Heliosphere up to 100AU

far from the Sun. These particlesare called Interplanetary medium.



/91

The solar wind explains why the second tail of comets(the plasma tail) always points away from the Sun.On Earth, solar winds can create continuous current onthe high-voltage power lines, creating overload of the

power transformers.The Earth’s magnetic field (magnetosphere) protects usmore or less against the solar wind. When they penetratenear the poles, they create the Aurora Borealis.

The plasma is ejected from the Sun at a speed of on average 450 km/s (between 400 –800 km/s) and is composed of 73% of hydrogen and 25% helium, roughly 10^6 T persecond.

Because the Sun is rotating, the magnetic field lines form aspiral, called Parker’s spiral.

13.4. Age of the Solar system

• Oldest rocks on Earth are 4.4bn years (“Gy”)• Oldest rocks on the Moon are 4.4 - 4.5bn years.• Oldest meteorite is 4.54bn years

Our best estimate is 4.55-4.58bn years, using radioactive dating.

Atom of atomic number N and of atomic mass A has its nucleus which contains Nprotons (positive) and A-N neutrons (neutral). A indicates the number of nucleons

Nuclei can have same N but different A. They are then called isotopes.



/91

13.5. Nuclear/radioactivity decay

Most combinations are unstable and decay via:• ! decay: emission of Helium nucleus, i.e. 2 protons and 2

neutrons

• & decay: emission of electron with conversion of neutron $ proton, or emission of a positron with conversion of proton $

neutron: or

• Fission: breakup into two smaller nuclei All these decays are usually accompanied by creation of ' rays, and produce heat.

Using Ec = % mv2 = 3/2 kB T (thermal velocity), we find that vhelium "vescape and thereforethe Earth is loosing Helium. But it is producing Helium also, through ! decay.

13.6. Radiometric datingWe use decay process, for example using Carbon 14:If for an atom, we know the half-life period t% (i.e. the period in which half of theremaining atoms have decayed), then the number of atoms remaining (i.e. that havenot yet decayed) is:

where N(t) is the number of atoms remaining, and t% the

half-life period.

Which means that every half-time period, half the atoms have decayed. Since atoms

have no memory, after the half-time period the process starts again.

Use ln to solve.

Carbon 14 dating can estimate a date from a few hundreds year to 50,000 years.

13.7. Radiometric dating (Uranium-lead)

This method can estimate a date from 1M years to 4.5bn years, with 0.1-1% precision,and uses the fact that Uranium decays into lead through 2 routes instead of 1. Hence

the 2 routes should give the same dating, which in practice is not the case. Hence wereduce uncertainty.1.

238U to

206Pb, and

2. 235U to

207Pb

Under conditions where the system has remained closed, and therefore no lead loss

has occurred, the age of the zircon can be calculated independently from the two

equations:



/91

And

In practice, the results of both equations differ slightly, because Fission tracks and

micro-cracks within the crystal will create conduits deep within the crystal, thereby

providing a method of transport to facilitate

the leaching of Pb isotopes from the zircon

crystal.

We use the upper intercept of the

Concordia method to evaluate the age of

the sample.

13.8. The creation of the solar system

Conditions for a molecular cloud to stay a cloud

Recalling that Ec = % mv2 = 3/2 kB T $ v=f1(T)

And that v2esc =G.M.m/R $ vesc =f2(R)

The cloud will remain stable if f1(T) = f2(R), i.e.

R=[G.M] / [3 kB T].

This can be also expressed as:

Ec < EG $ 3/2 kB T< G.M.m/R Jeans instability

Where M is the mass of the cloud, and m the mass of

each particle. Hence, if clusters form, temperature gets higher and creates stars.

In practice, if there’s a singularity, or a non-conformity, the cloud will split.

The solar system started as a molecular cloud. Then fragmentation and gravitational

collapse created by a nearby supernova creates a fragment of around 3000 MassSun,

and 2000-20000 AU in size, from which the Sun is a member of an open cluster now

dispersed.

If the mass had a small rotation movement, when it collapses the mass concentrates in

one point, and therefore the rotation speed increases (conservation of the angular



/91

momentum). That is why our galaxy, and all stars and planet, orbit together in the same

direction. The closer the planet to the axis, the faster is moves.

Matter orbiting around the center flattens to a ring

towards the center of the rotation, exactly as what

happens we turn around quickly, holding strings,which move towards our center of orbit.

13.9. Kelvin–Helmholtzcontraction: gravitycontraction! heat

As the universe concentrates, heat increases. This is also true for a star or planets.

This is due to the law of conservation of energy, where . If d (, then

energy must be radiated through temperature increase.

At the center of our galaxy, temperature "2000

K, with highest planet density. Then the farther

the planet from the center, the colder it gets, and

the more solid we can find on the planet.

13.10. Terrestrial formation

1. Grains of dust collide and adhere2. As soon as they reach 1km in size, they are bound by gravitation

3. The larger the object, the faster it grows. The growth rate is proportional to R 4

4. Objects grow for 100,000 years, at which point they are called protoplanets (R %

1000km)

5. Because of the Kelvin–Helmholtz contraction effects, the gravitational force heats the

elements until the planet melts. The planet becomes spherical because 1) it melts and 2)of gravity

6. Chemical differentiation occurs: heavier materials sink to core7. Gravity is opposed by pressure force: pressure increases when closer to the center

8. Compression heats the core

9. Then, protoplanets accrete into larger planets, called planetesimals. It ends up with 100Moon-Mars sized planetesimals.



/91

10. Gravitational interactions distorts orbits

11. Collisions lead to merger or ejection, leaving large Venus or Mars. Other get stripped to

core

12. Orbits settle to near-circular orbits in 10-100 Myears, around the Sun

13.11. Beyond the Snow lineThe snow line is the distance from the Sun where it is cool enough for hydrogen compounds

such as water, ammonia and methane to condense into solid. The temperature is estimated

around 150K. The snow line of the solar system is around 5AU, hence Jupiter is right on the

outside of this line.1. Outside of this line, the gravity of the Sun is lower, and gas giants like Jupiter acted like

our solar system: it aggregated nearly all the remaining H and He gas, and grew veryrapidly until gas in orbit exhausted.

2. Jupiter rotates, exactly like our solar system, but faster (in 10h)3. Jupiter creates a flatten ring, exactly as the Sun creates its planets orbiting

Saturn is further from Sun, it started later so captured less gas, but repeated the same process.

13.12. Orbital resonance

Orbital resonance occurs when planets’ orbital periods the Sun are

related by a ratio of small integers: nP1 = mP2.

• Orbital resonances greatly enhance the mutual gravitational

influence of the bodies, i.e., their ability to alter or constrain each

other's orbits, regardless of the Sun’s attraction.

• Often, resonance occurring at 2-4 AU disrupts planet formation,

which creates asteroid belts around the planet. This is the case in

particular for Jupiter and Saturn.• In most cases, this results in an unstable interaction, in which the bodies exchange

momentum and shift orbits until the resonance no longer exists.

Kirkwood gapResonance can also occur between asteroids gravitating

around the Sun at a distance coming at resonance withJupiter’s orbit. At those distances, no asteroids can be found.

This is called the Kirkwood gap.

The Nice modelThe Nice model, developed at the Nice observatory, explains why the Gas giants are where they

are. It proposes the migration of the giant planets from an initial compact configuration into their present positions. The four model proposes that after the dissipation of the gas and dust of the

primordial Solar System disk, the four giant planets (Jupiter, Saturn, Uranus and Neptune) wereoriginally found on near-circular orbits between ~5.5 and ~17 astronomical units (AU), much

more closely spaced and more compact than in the present.



/91

After several hundreds of millions of years of slow, gradual migration, Jupiter and Saturn, the

two inmost giant planets, cross their mutual 1:2 mean-motion resonance. This resonance

increases their orbital eccentricities, destabilizing the entire planetary system. The arrangement

of the giant planets alters quickly and dramatically. Jupiter shifts Saturn out towards its present

position, and this relocation causes mutual gravitational encounters between Saturn and the two

ice giants, which propel Neptune and Uranus onto much more eccentric orbits.

These ice giants then plough into the planetesimal disk, scattering tens of thousands of planetesimals from their formerly stable orbits in the outer Solar System. This disruption almostentirely scatters the primordial disk, removing 99% of its mass, a scenario which explains the

modern-day absence of a dense trans-Neptunian population. Some of the planetesimals arethrown into the inner Solar System, producing a sudden influx of impacts on the terrestrial

planets: the Late Heavy Bombardment.

13.13. Timeline of the creation of the Solar system



/91

14. The Earth

14.1. General

• 71% of its surface is water

• Above this is atmosphere of mostly N2 and

O2

• Surface is rocky: Si

• The core is rich in metals: Fe, Ni. Inner

core is solid, outer core is liquid

• Mantle is made of rocks

• Average density 5500 kg/m3 (rock is 3000

kg/m3)

• Pressure, density, temperature increase with depth

14.2. Internal HeatHeat is generated in the core by :

• Radioactive decay

• Kelvin-Helmholtz

The mantle drives the convection: fluids go up, then down.

Because of that, the crust - broken into plates - is dragged

by mantle. This creates mountains at the surface. New crust

arises from volcanic processes.Heat loss: 87W/m2 at the surface.

However, most heat on Earth comes from the Sun’s radiation. It loses energy as radiation intospace.

14.3. Finding temperature of Earth

Ignoring Earth’s atmosphereEnergy captured on Earth (assuming Earth is black): Iin

and also (L is in W)

therefore

(T is of Sun)

Energy radiated by Earth (assuming Earth is black): Iout



/91

At equilibrium: Iin = Iout ! gives 279K, i.e. 6°C which is too cold!

In fact, Earth is blue and therefore reflects about 0.367 of the radiation, therefore Iabs = (1-a) Iin

where a = Earth’s albedo = 0.367 ! gives 248K which is even

colder (and below zero)!

TEARTH is the temperature of the planet based only on the light received by the Star.

Greenhouse modelGreenhouse effect explains why temperature is higher:

• The atmosphere is transparent to the incoming sunlight (visible)

• The atmosphere partially (g) absorbs the infrared light radiated by Earth, through it’s

molecules, reradiating part of this energy towards Earth.

At equilibrium, each medium is such as Fin=Fout.

• Atmosphere: g., Te4 = 2., Ta

4

• Earth’s Surface : , Te4= , Ta

4+ Fin

Solving the equations gives

Fin = (1-g/2). , Te4

Te = (1-g/2)-1/4 Tno greenhouse

This gives: a=0.367 and g=0.21 ! Te = 292K

a depends on clouds, g depends on molecules present in the atmosphere.

Notice that changes in a & g can alter climate drastically.

14.4. The Atmosphere

N2 and CO2 were released when minerals werecooked at high temperature.

H2O was imported from outer system as ice, duringheaving bombardment period (3.5bn years ago)

Rain creates oceans which dissolve CO2 and fix it in

sediments

Plants released O2 initially taken up by Fe and S



/91

14.5. Earth magnetism

Earth is a magnet, roughly aligned with rotation axis.

1. The core – metal-rich - is being heated by the inner code. This drives convection.

2. Along with the rotation of the Earth, this creates a magnetic field which aligns itself with

the rotation axis

3. Every 500 years, the N/S change polarity unpredictably

Charged particles of Solar wind are trapped by field lines into radiation belts. This prevents thisintense flux of charged particle to arrive on Earth.

The solar winds deforms the Earth’s field, in particular on the North pole where they penetrate

Earth’s atmosphere. This gives the auroras in the poles.



/91

15. The Moon

Those linear marks indicate that the Moon is shrinking.

• Temperature 370K day, 100K night (nights/days are 2 weeks long)

• No liquid water because water requires atmospheric pressure to retain it. Ice in crater

shadows 35K

• No atmosphere because the Moon is not big enough to attract the molecules

• Lunar surface is a museum of history, because on Earth those marks are removed by

tectonic activities

• Moonquakes are caused by Earth’s tidal forces

• No big magnetic field

• Mineral composition indicates it's a peace of Earth. Probably from a giant impact by a

Mars-sized meteorite in early Earth history. This explains why Moon is poor in metals,since metals are in the Earth's core,



/91

Combining crater dating

with radiometric dating of

lunar samples and

meteorite leads to history

of bombardment rates,and 3.9bn years ago the

period of heavy

bombardment.

• Lunar density is not much higher than that of rocks,

therefore we deduce its core is very small.



/91

16. Planets detection methods

16.1. Astrometry

First other extra solar planets discovered in 1988. There are now 853 planets found, around

672 stars. Only 32 planets, in 28 systems have been detected by imaging.

In a system with 1 planet orbiting a star, the star is slightly

orbiting around the center of gravity of the system. We have:

Mstar .R star = M planet.R planet

R system = R star + R planet is fixed

By combining both equations, we get:

R star = (M planet /Msystem).R system

By carefully analyzing the complex path of a Star, we can get information on orbiting

planets.

16.2. Radical velocity

We can measure the Star’s speed using Doppler effect:+ = + 0 (1-v/c)

This is valid of course if the planet is in our plan, because

to measure the Doppler effect, the planet must goaway/closer from us.

Forcegravity, system = Forcecentripetal, star

Which gives using R star = (M planet /Msystem).R system :

V planet

498 planets in 386 systems have been detected by radial velocity measurements.

16.3. Transit method

If a planet crosses (transits) in front of its parent star's disk, then the observed visual brightnessof the star drops a small amount. The amount the star dims depends on the relative sizes of the

star and the planet. For example, in the case of HD 209458, the star dims 1.7%.This method has two major disadvantages.

The farther the planet from the Star, the more

effect it has on its velocity.

Jupiter’s effect on the Sun is 12.5 m/s.



/91

• First of all, planetary transits are only observable for planets whose orbits happen to be

perfectly aligned from the astronomers' vantage point. The probability of a planetary

orbital plane being directly on the line-of-sight to a star is the ratio of the diameter of the

star to the diameter of the orbit. About 10% of planets with small orbits have such

alignment, and the fraction decreases for planets with larger orbits. For a planet orbiting a

sun-sized star at 1 AU, the probability of a random alignment producing a transit is

0.47%. Therefore the method cannot answer the question of whether any particular star isa host to planets.

• Secondly, the method suffers from a high rate of false detections. A transit detection

requires additional confirmation, typically from the radial-velocity method.

However, by scanning large areas of the sky containing thousands or even hundreds of thousandsof stars at once, transit surveys can in principle find extrasolar planets at a rate that could

potentially exceed that of the radial-velocity method.The main advantage of the transit method is that the size of the planet can be determined from

the lightcurve. When combined with the radial-velocity method (which determines the planet's

mass) one can determine the density of the planet, and hence learn something about the planet's

physical structure.The transit method also makes it possible to study the atmosphere of the transiting planet. When

the planet transits the star, light from the star passes through the upper atmosphere of the planet.

By studying the high-resolution stellar spectrum carefully, one can detect elements present in the

planet's atmosphere.

290 planets in 235 systems have been detected via transit. The Kepler telescope has found 2321

candidate planets in 1290 systems.

16.4. What have we found?• Between 1-40% of (Sunlike) stars

have planets

• Only a tiny zone has been explored.

• These methods are sensitive to HotJupiter: big planets orbiting close to

stars.



/91

17. Analyzing Stars

17.1. Laws of conservations of charge and of electrons

1. Law of conservation of charges: Whatever the reaction, the total charge remains

unchanged before/after the reaction has occurred.

2. Law of conservation of electrons: Whatever the reaction, the number of electrons

remains unchanged before/after the reaction has occurred.

17.2. Chemical reactions to create heat

The Sun gets part of its heat through chemical reactions. It burns 10-19 J per atom, or 6.107 J

per kg of H.

The Sun produces in this fashion 6.4.1018 kg/s, hence the Sun would live around 10000 years

if using this process only.

17.3. Nuclear Fission to create heat

Reminder:

1 atom = 1 nucleus + electrons orbiting

= a bunch of nucleons + electrons orbiting

Recall that nucleon = neutron or proton. Atoms are not charged, because the charges of thenucleons and the electrons cancel each others.

Why don’t nuclei break up under electric repulsion? A strong, short-range (10-15m) attractiveforce binds the nucleons. This gravity force is called the nuclear force.

If we can break the nucleus, then the nucleons get away from each other, and liberate

electrostatic energy that was used to bind them together.

Practically, the binding energy per nucleon peaks around iron (Fe).

Same process happens for decay.



/91

17.4. Nuclear fusion to create heat: PP chain

Two atoms of H can crash and merge if 1) they have enough kinetic energy to overcome

electrostatic repulsion, and 2) right alignment.

1. When they merge, the 2 protons create a

proton and a neutron. Because of the law

on conservation of charges, one positiveelectron (positron) gets created. Because ofthe law of conservation of the quantity of

electrons, a neutrino ( -) gets created.1H +

1H $

2H + e

+ + -e + 0,42 MeV

(2H is also called deuterium)

2. The positron e+ positron annihilates itself with

an electron of a nearby H atom, which creates

energy through 2 photons

e+ + e

- $ 2' + 1,02 MeV

3. The deuterium can then repeat the same

process with another H atom2H +

1H $

3He + . + 5,49 MeV

4. Two3He will eventually merge

3He +

3He $

4He +

1H +

1H + 12,86 MeV PP1

This process creates 4.3 10-12J through 1 He atom,

guaranteeing the Sun ~ 1011

years (100 bnyears)

However, this process happens very infrequently (1 in 5bn years), and only temperature in the

core is high enough for this process to occur in the Sun, which represents 10% of the Sun. Hence

this fusion process guarantees 10bn years. Which is also roughly the Sun’s life, by the way.

Exercise: quantity of He produced since the Sun’s birthEnergy produced since birth: Luminosity (in J/s) x Sun’s life in s = 3.83e26 x 1.4e17s = 5.4e43 J

Energy produced by 1 He atom in PP chain: 4.3e-12 J/atom of He

Mass of 1 He atom ! 4 protons = 6.65e-27kg/atom

Therefore; 8.41e28kg of He have been produced in the core (on top of the 27% already present

uniformly).



/91

17.6. Solar weather

• The Sun is covered by sporadic sunspots, varying in 1 year cycle.

• Looking at those sunspots, we can see that the Sun is orbiting in

~ 25 days.

• Sunspots pair (+/-) appear first at mid-latitudes, and later near

equator.• Spots are regions of increased magnetic fields, therefore they

modify the density of atoms and therefore temperature. The

charge varies from one cycle to the next, because the magnetic field reverses between

cycles.

• The equator rotates faster than the poles! This deforms the

convection zone: the field gets elongated in the equator vs. the poles. Reconnection releases energy, every 11 years reverses

polarity.

• Sunspots are those regions where the magnetic field rotates when

getting released

• Reconnection releases magnetic energy through charged particles: up to 6x1025

J, in gas

@107K. This is the solar wind, which takes ~3 days to arrive to Earth, if projected in our

direction



/91

17.7. Parallax, or Finding distance from the Star

To find the distance from a Star to Earth, we look at the angle at which we see the Star, from 2

distance points.

Dsun/D = tan(1) ! 1 = Atan(Dsun/D) % Dsun/D

Atan(r) % r if r <<1, i.e. typically expressed in radians (1° NOK, but it gives 0.02 radians OK)

Tan(r) % r if r <<1

where DAU is D expressed in AU

where

Evaluating the distance of Stars enable to build a 3D map of our universe. Today, we have

registered 2.5M stars. Gaia mission will be launched in 2013 to extend our knowledge.

Parallax formula above is used when angle is parallax angle (i.e. angle between 2 Earth’s

positions). This is NOT to be confused with angle we see from one fixed position on Earth! In

this latter case, we must use angle x Distance from Earth, where angle is in radians.

Convert to

Convert to arc secondsConvert to arc seconds

206265 is the AU-

to-parsec conversion

factor



/91

17.8. Astrometry, or finding the speed of the Stars

Stars move, very slowly

Over the course of centuries, stars appear to maintain nearly fixed positions with respect to each

other, so that they form the same constellations over historical time. Ursa Major, for example,

looks nearly the same now as it did hundreds of years ago. However, precise long-term

observations show that the constellations change shape, albeit very slowly, and that each star hasan independent motion.

Defining motion

The proper motion of a star is its angular change in position over time as seen from the centerof mass of the solar system. It is measured in seconds of arc per year, arcsec/yr, where 3600

arcseconds equal one degree. This contrasts with radial velocity, which is the time rate of changein distance toward or away from the viewer, usually measured by Doppler shift of received

radiation. The proper motion is not entirely "proper" (that is, intrinsic to the star) because itincludes a component due to the motion of the solar system itself.

Radial velocity is computed using Doppler effect:

VR = c.(+ /+ 0 – 1) + is the observed wavelength of

a known atom, + 0 the real wavelength at v=0

Transverse velocity is computing using:

VT = 4740.µ. D

Where VT is in m/s

µ is in arcsec/yearD in parsec

1AU/(365.25x24x3600)% 4740 Space velocity 2 = VR 2 + VT

2

Notice that Vr points out of the Sun, i.e. by convention in the above formula a planet going

farther from us has Vr>0 (opposite sign of the one used in the Doppler effect)

By studying the star’s spectrum of radiation, spectroscopy can also derive many properties of

distant stars and galaxies, such as their chemical composition, temperature, density, mass,

distance, luminosity, and relative motion using Doppler shift measurements.



/91

17.12. Recap: Analyzing Stars

By visual observation

• From relative brightness we deduce Luminosity: L= 4 & D2 b !

A brighter star is more luminous, if at the same distance from Earth (to be checked with

parallax).

• From color (spectrum) we deduce Temperature A blue star is hotter than an orange star.

• From the luminosity and the temperature we deduce the Star’s radius:

!

• From the radius of the star 1, and the orbiting period of a nearby planet/star 2, we can

find the star’s mass: where a (orbiting radius) is in AU and P in years

• From the mass of the star, we can find its orbiting radius relative to the center of mass of

the system by solving:

M1 . R 1 = M2.R 2 M1 + M2 = a3/P2.Msun

• From M and R we can deduce surface gravity:

• Form speed of Star we deduce Mass around which it I orbiting:

Using Doppler effect• We can find the radial velocity VR = c.(+ /+ 0 – 1) if star

moving in our direction

• Using VR1,2 and observing P, we can deduce the orbiting

radius a1,2 = VR1,2.P/2&

• We can also deduce the ratio of the masses: M1. R 2 = M2.R 2

! M2 / M1 = VR1,2/ VR2,1! Msystem = M1 .(V2 + V1) V2 and M1 . VR1,2 = M2.VR2,1

• R = R 1 + R 2 because orbiting planets/stars are always in opposite direction

R = P.V1/2& + P.V2/2 & ! R = (V1 + V2) .P/2&

where P is in year, and Vearth = 29.78km/s speed of Earth orbiting

the Sun. V are the radial velocities.



/91

Particular case of Eclipsing binaries

To compute dip in intensity, do not use b = "max.T, because

"max are not the same for each star since speeds are not the

same. Instead, use L # b. D2 and L # R

2T

4

(V1 + V2). t2 = 2. R 2 t2 is time of slopes. Notice it’s the same for each dip, because S2 iseither going in front, or behind S1

(V1 + V2). t1 = 2. R 1 t1 is defined as the eclipse time. By definition, eclipse occurs when

slope starts to decrease to lower brightness, until when it startsincreasing.

(V1+V2).t3=2.(R1-R2) t3 is the time of the slope when decreasing to lowest brightness.

17.13. Mass-luminosity relation

Broadly speaking, the heavier the star, the hotter, and the more luminous, and the shorterits life.

“You give Nature a bowl of hydrogen, and it will produce a Star”.

3.5 applies to main-sequence stars (2.Msun < M < 10.Msun)

This makes sense, because the bigger the star, the higher the density at the core, the higher the

nuclear reactions, the higher the luminosity. This means big stars run out of H must faster.

17.14. Main-sequence starsThis is our Sun

• Main-sequence stars are basically all identical to our Sun, but with different sizes.

• 85% of Stars are main-sequence stars

• Main-sequence stars fuse Hydrogen to Helium in core

CNO cycle for MS stars > 1.3 Rsun • In star with R>1.3 R sun, temperature is high enough

to allow another chain or reaction: the CNO cycle.Carbon is used (but not consumed) as a catalyst to

produce much more energy. This is because a protonand a carbon nucleus crash into each other (vs. a

proton-proton in the PP chain), which requires muchmore energy since repulsion is much higher by the more-

charged C.



/91

Radiation and Convection effects

Expansion by contractionH in the core produces He through fusion, and more and more He get produced in the core, and

less and less H. Hence, probability of collision of protons decreases. Hence, the rate of fusiondecreases in the core, as well as the radiation pressure. Because 4 H ! 1 He, the pressure should

decrease as well. But this is not possible! Because the outer layer pressures the core, and the core

contracts. Hence pressure increase, and density increases. This process is called Expansion by

contraction. A new influx of H arrives from outer layers. Net result:

1. The core contract and heats

2. Luminosity increases

3. Envelope expands

For example, our Sun is 25% brighter now than when it formed, hence Earth was colder. Its core

is now 60% He. Therefore, in 1-3 Gyears, Earth could become uninhabitable. But other effects

are in play: more solar wind, hence Earth farther way from Sun.



/91

18.3. Main Sequence

For 10.9G years, the star has sat happily within the same point of the HR diagram. At the end,

R $1.58 R sun, L$2.21 Lsun. T $ 10.9G years.

• Hydrogen fusion in the core supports envelope by thermal and radiation pressure

• Luminosity and surface temperature are determined mainly by mass, but also bycomposition, rotation, close binary partner, atmospheric and interstellar effect

• Over time, the core contracts and heats & fusion rate increases + envelope expands

slowly with little change in temperature. Intert He gets deposited into the core, thereforecore is increasing. Luminosity is now coming from H shell around the He core.

Temperature remains roughly the same, while R is increasing, hence L increases.

18.4. From MS to Subgiant stars

Let's consider what's happening at the end of Sun's MS («post MS»). This period of Subgiant star

lasts until T $ 11.6 G years. R $2.3R sun, L$2.2Lsun.

P.V = N. k B.T therefore if V' (because of enveloppe expanding), P(, but there is a limit until

which pressure in the core can decrease and still support outer layers:

When core becomes too large (Mcore ) 8% M), it cannot support outer layers and collapses

rapidly, making it a Subgiant star.

This creates Gravitational energy which expands the enveloppe. Temperature decreases.

A Subgiant star has a state life of a few hundreds of Myears. It then becomes a red giant star.

SummaryCore becomes too large * Pressure too small * core collapses under outer layers *

Gravitational energy * envelope expands * T°C (

18.1. From Subgiant to Red giant branch stars

For our Sun, the period of Red giant star lasts until T $ 12.233 G years. R $ 166 R sun, L$ 2350

Lsun.

The core collapses, pression heats shell which increases Luminosity. CNO cycle occurs (Carbon

acts as catalyst).

The enveloppe expands, heat is radiated outside through convection. The Star looses up to 28%

of its mass through Solar wind.

And then?

The core does not collapse due to electron degeneracy: Pe = K e +5/3

, where K e $ 3.2 106 Nm

-2

for the Sun, K e =2.35684210338

/µ5/3

otherwise, where µ the average mass per free electron. This

pressure prevents the core from collapsing.

Electron degeneracy means that up to a certain contraction, the electrons cannot be packed

further (Pauli principle).

Summary

Core collapse * pression heats shell * Luminosity' and CNO* * heat radiated outside *

mass evacuated through solar wind

*: Carbon acts as catalyst



/91

18.2. From Red giant branch to Helium core flash

• When the temperature of the core reaches 108 K, 4He and 4He fuse (the process is called

triple-# process), create a 8Be atome with emission of

gamma particle, and8Be fuse again with

4He atom to

create a 12C atome and emission of a gamma particle

through explosions. The core degenerates into12

C.

• For a few seconds, the star produce a galactic luminosity

(hundreds of k of Lsun), called Helium Flash, absorbed bythe star's atmosphere, ejecting part of the atmosphere and

leading to mass loss.

• The amosphere increases due to this flash, therefore the

Luminosity decreases. This contraction creates heat, therefore temperature increases.

Summary

High T°C * triple-# process* * Helium flash * enveloppe expands then contracts * T°C '.

*:

4

He fuse + creates

8

Be which fuses and creates

12

C

18.3. From Helium core flash to Horizontal branch

T $ 12.234 G years. R $ 10 R sun, L$ 41 Lsun.

• Core is contracting, creating heat

• Envelope is contracting and heating as well

• Shell is fusing H to He

Summary

Core & Envelope contracting * T°C '. Core & Envelope contracting + T°C ' * Luminosity'.



/91

• Because core is degenerated, pressure in the core for degenerated electrons must equal

gravity pressure: K e.+5/3 ! #. M2/R 4 * MR 3 = cst * in white dwarves, the higher the mass,

the smaller the star. In fact, it can be shown by relativity that there's a limit: Mdwarf <1.44Msun

(Chandrasekhar's limit)

Summary Hot, degenerated CO core, cooling down and surrounded by its gas nebula. Because degeneratedelectrons support gravity pressure * Mdwarf <1.44Msun . R dwarf & 0 as Mdwarf & 1.44Msun

18.7. White Dwarf Nova

• A white dwarf can transfer mass as H from a nearby Giant as binary partner, at a rate of up to10-8 Msun/year.

• The coming H gas is heated by gravity, this increases temperature.

• When 10-8 Msun accumulated, temperature reaches 107 K, and CNO fusion occurs.Temperature reaches 10

8K, Luminosity 10

5Lsun

• CNO heating is done explosively: at 108K, the radiation pressure overcome the electron

degeneracy pressure, and material is ejected: 1038J over a few months.

• Mass reaccumulates, so this process can occur every 105 years

• Ejected matter glows at 9000 K

• 30 White Dwarf Nova are observed per year in M31 galaxy

Summary

Giant binary partner transfer H mass to White Dwarf * H gets heated by gravity * T°C' *

CNO heating * huge radiation & Luminosity & material ejected. Recurring process.

18.8. Supernova (type Ia)

• If the mass transfer to a white dwarf gets closer to the Chandrasekhar limit (1.44 M sun),

the degenerated C in the core fuses. C is degenerated, therefore heating does not lead to

expansion but to explosion: shock waves blow star appart, ejecting matter at high speed

and releasing 1044J

• Temperature exceeds 109K, Luminosity reaches 10

-10Lsun over a few months

• Spectrum has absorption lines of Si, but little H and He

• Mass donor is either MS/Giant, or more often another white dwarf ripped apart by tidalforces in merger.



/91

If He is at the right temperature,

compression of the star ionizes it, so less

energy is spent on heating during

contraction. The layer becomes opaque,

capturing contraction energy. When it'scontracted enough, it releases the storedenergy, and expands (reducing ionization).

This process is called k mechanism.

Summary Variable stars pusle, and are stars successively contracting and expanding. T°C and Luminosity

peak during expansion. They can be used as standard candles because Luminosity is function of

period (Log(L) = # Log(Period)), and with measured brightness we find cluster distance.

18.10. Blue stragglers

How can we explain why some massive stars are still in the MS sequence, when they should not

be (more massive stars evolving faster)? This can be explained by mass transfer: a lighter starhas attracted matter ejected by a more massive binary partner, who was becoming a subgiant. As

a result, the lighter star has become the most massive, while the other has become the lightest.

18.11. From MS to Red Supergiant• Stars whose M > 8 Msun end main

sequence in 10M years. R ! 5 AU

• Burning shell is H, core is inert He

• When fusion H stops, core contracts and

envelope expands and cools

Summary Fusion H stops early because massive star, core contracts and envelope expands and cools.

18.12. From Red Supergiantto Helium flash to Blue

Supergiant• He core ignites

• H fusion in shell



/91

• Envelope contracts and heats, forming CO core

• Temperature increases, therefore star becomes «blue»

Summary

He core ignites * triple-# process* * Helium flash * enveloppe expands then contracts *

T°C '.

*: 4He fuse + creates 8Be which fuses and creates 12C

18.13. From Blue Supergiant to Massive star AGB

• CO core collapses until T > 6. 108K

• Then C fuses, producing Mg Ne O

• Superwind and mass loss: nebulae

• If star is big enough, when T > 1.5 109K Ne fuses and produces O Mg. Neutrinos carry

off L = Lsun

• If star is big enough, O fuses when T > 2.1 109K and produces Si S P. Neutrinos carry off

L = 105Lsun, lasts 1 year

• Si fusion occurs at T > 3.5 109K and produces

Ni Fe. Neutrinos carry off L = 1012Lsun, lasts 1

day

• S-process nucleosynthesis then producesheavier elements. Fe is the end.

• Core radius = R earth, Envelope radius ! 5 AU

Summary

T°C ' * heavier elements get created, fuse * T°C ', and create more heavier elements: Mg,

Ne, O, Si, S, P, Ni, Fe



/91

18.17. From Supernova type-Ib/Ic/II to Neutron (pulsar) star

• Core of supernova type Ib/Ic/II stops collapsing due to neutron degeneracy, at + ! 7.1017

kg/m3

• Surface gravity 1.9 1011

g

• Mneutron star = 1.4 Msun and R neutron star ! 10 km

• MR 3 = constant

• Rapid rotation due to conservation of angular momentum (R becomes smaller, hence

rotation speed increases). Using MvR = MR 2/P we get

where Pcore is the rotation period of the core before collapse

• That gives Pneutron star !0.005s. That's 200 rotations per second!

• Maximum mass: Mchandrasekhar = 2.2 Msun – 2.9 Msun depending on rotation

• Same for magnetic field: very high magnetic field because everything is ionized

where Bcore is the

rotation period of the core before collapse

• That gives Bneutron star !1012

Bsun

• Temperature decreasing, but still high: T !106K

• "max !3.9 nm

• L ! 0.25 Lsun

• Rapid rotation creates regular pulse, hence

neutron star pulse: 0.2s % P % 2s

• This rotation time is very regular, nearly as precisice as atomic clock. But slowly decreasinganyway, gone in 107years

• Minimum rotation period, after which star blows

appart:

That means that if star rotates faster than this, centripedal forces are higher than gravitational

forces and star blows appart. This comes from v2/R (=4-2R/P2) = GM/R 2

• Pulsar emit at all wavelength bands

Summary

Supernova stops at electron degeneracy. If mass big enough * core contracts further and nearly

all electrons are transformed into neutrons, core collapse stopping at neutron degeneracy.

Rotation is very rapid, creating beam of neutrons at regular pulse.



/91

18.18. Recap: Stars on HR diagram

10 y

109 y

10 y10

7

y



/91

19. Relativity

19.1. Principle of Relativity

• Laws of physics are the same measured at rest or at constant

velocity

• • Speed of light is the same whatever the referential

• “At rest” is meaningless. Only relative velocities are physical

19.2. Spacetime

• Space:

• Space motion:

•

Spacetime: all possible events (t,r) = (t,x,y,z)• Worldline: path of an object as it travels in spacetime

•

19.3. Lorentz transformations

x’ is x seen by Observator

y’=y, z’=z

By setting:

x' = Ax + Bt

t’ = Cx + Dt

And using:

• For x’=0, x=v.t

• For x=0, x’=-v.t’

• c is constant for both

• A(v) = A(-v)

• solving x’ and t’ in function of x and t, and then rewriting the symmetric equivalent

We find: A(-v) = 1/(A(v) (1-v2/c2))



/91

19.4. Relativistic Spacetime

From relativity equations, we deduce that the same even, happening 1 light-year away, appear at

! 1sec difference for 2 observers: 1 standing still, the other driving at 10 m per sec.

19.5. Length contraction

It follows from the Lorentz transformations that lengths contract. Seen from an observer standingstill, the length of an object moving relatively to this observer with a speed v appears to have a

length of

Notice that this equation is symmetric (same value for v or –v).

This is the length seen by the other observer.

19.6. Time dilation

Each observer thinks the other’s clock is ticking slower!

For example, if we observe a particle is going at 0.8c, the particle itself sees time T’. This isimportant for particle decay.

19.7. Doppler effect due to high speed

Red shift

Blue shift

19.8. Velocity addition

19.9. Lorentz metric

The unification of space and time is exemplified by the common practice of selecting a metric

(the measure that specifies the interval between two events in spacetime) such that all four

dimensions are measured in terms of units of distance: representing an event as (ct,x,y,z) in the

Lorentz metric.

In the (ct,x,y,z) spacetime, (ct,x,y,z) represents an event, which is characterized in terms of space

AND time.



/91

19.10. The Invariant Interval

The separation between 2 events is measured by the invariant interval, s2, between 2 events. Note

that the interval takes into account not only the spatial separation between the events, but also

their temporal separation.

s2 = c2.(#t)2 - (#r)2 Other conventions put s2 = (#r)2 - c2.(#t)2

The measurement of lengths is more complicated in the theory of relativity than in classical

mechanics. In classical mechanics, lengths are measured based on the assumption that the

locations of all points involved are measured simultaneously. But in the theory of relativity, the

notion of simultaneity is dependent on the observer. Proper distance provide an invariant

measure, whose value is the same for all observers.

In relativity, proper time is the elapsed time between two events as measured by a clock that

passes through both events . The proper time depends not only on the events but also on the

motion of the clock between the events. An accelerated clock will measure a smaller elapsed

time between two events than that measured by a non-accelerated (inertial) clock between the

same two events. The twin paradox is an example of this effect.

Proper time simply means classical, non-relativistic time, but taking into account time dilatation

due to relativity. Notice indeed that

Proper distance is analogous to proper time. The difference is that proper length is the invariant

interval of a spacelike path or pair of spacelike-separated events, while proper time is the

invariant interval of a timelike path or pair of timelike-separated events.

Time-like interval

For two events separated by a time-like interval, enough time passes between them for there to

be a cause–effect relationship between the two events. For a particle traveling through space at

less than the speed of light, any two events which occur to or by the particle must be separated

by a time-like interval.



/91

Spaceship in space, no gravity, acceleratingwith acceleration g.

Spaceship on the ground, gravity g.

In both cases, the trajectory of the ball is the same! This means that the gravitational force is

simply an acceleration.

As an object in free-fall approaches the ground, the time scale stretches at an accelerated rate,giving the appearance that it is accelerating towards the planetary object when, in fact, the falling

body really isn't accelerating at all. This becomes evident when ones realizes that someone infree fall does not feel any acceleration! This is why an accelerometer in free-fall doesn't register

any acceleration; there isn't any.

19.14. Gravitational redshift

Gravitational redshift: The wavelength of a photon (e.g. light, or any

electromagnetic radiation) as seen by an observer in higher

gravitational potential increases compared to its wavelength seen at the point of emission. This is equivalent to saying that its frequency

decreases towards the red part of the light spectrum.

Gravitational blueshift: The wavelength decreases, and frequency

increases towards the blue part of the light spectrum, as seen by anobserver in weaker gravitational potential.

The Gravitational shift is where + e is the wavelength of

the radiation measured at the source of emission, + e is the wavelength measured by the observer.

Therefore, in binary stars, we will observer a change in the Doppler shift of a star depending on

whether the other star is behind or before it, because the partner’s gravity affects the redshift.OBSERVED FROM GREAT DISTANCE

NEAR EARTH, at distance H

R is the distance between the center of mass of the gravitating body, and the point at which the

proton is emitted.



/91

is called Schwarzschild radius : the distance from the center of an object such that, if

all the mass of the object were compressed within that sphere, the escape speed from the surfacewould equal the speed of light.

The redshift is not the Doppler shift: both objects are moving. And unlike Doppler shift, the

redshift is symmetric.

19.15. Relativistic Potential energy

Mass of the system (Sun + Earth) is slightly less that the sum of their masses, because of the

negative potential binding Energy between the two. Therefore, gravitation fields are nonlinear.

The relativistic version becomes:

In close binaries, v is typically very high, therefore we could use the relativistic version of the

potential energy to get better approximation.

19.16. Gravitational lensing

Relativity theory predicts light being deflected by planets by an angle :

This effect creates duplica or artefacts when looking at the stars: farther

galaxies may seem duplicated by cluster.

The deformation of spacetime around a massive object causes light rays to be deflected much

like light passing through an optic lens.

Relativistic

Potential Energy

Kinetic Energy



/91

19.17. Gravity is geometry

Gravity deforms spacetime, provoking attraction. That’s how gravity attracts objects.

A very good way of seeing it is trough dark matter: dark matter has mass, but does not interact a

lot with matter. So why does it get attracted by massive objects?

This is because massive object change spacetime, which attracts dark matter by pulling it closer.

This illustrates that gravity is geometry. It has nothing to do with the fact that dark matter does

not interact much with baryonic matter.

Dark matter is also gravitating around the center of galaxy, as are our sun and other stars/planets.

19.18. Gravitational waves

As massive objects move around in spacetime, the curvature changes to reflect the changedlocations of those objects. Moving objects generate a disturbance in spacetime which spreads

like electromagnetic waves. This disturbance is called gravitational wave. According to generalrelativity, gravitational waves travel through the universe at the speed of light. Gravitational

waves cannot exist in the Newtonian theory of gravitation, since in it physical interactions propagate at infinite speed.

Sources of detectable gravitational waves could possibly include binary star systems composed

of white dwarfs, neutron stars, or black holes.

In addition, binaries lose energy through Gravitational waves, therefore their orbits will decrease.



/91

20. Black holes

20.1. Horizon

For objects reaching the Schwarzschild radius (called the horizon of the black hole), the redshift

becomes infinite: a black hole. No light can emerge. Inversely, blueshift is observed from a blackhole.

For object of 1 solar mass, Schwarzschild radius R s = 3km.

Observed from far away, something falling in the black hole will actually never gets there: it willapproach, slowing down forever because of time dilation. The object becomes dimmer forever.

• Smaller stable orbit for a black hole is 3 R s.

• Photosphere at 1.5 R s : light circles around the black hole in unstable orbit.

• Tidal forces can become extreme, depending on Black hole’s mass:

since

Therefore, the more massive the Black hole, the smaller the tidal force ! So if the mass of the

black hole is very big, we may not notice it.

20.2. Singularity

Singularity of the back hole is where tidal force

becomes infinite. Even matter gets dislocated, and

Relativity equations break down – we do not know

what’s happening there.

20.3. Emission of X-rays

In the case of compact objects such as white dwarfs, neutron stars, and black holes, the gas in the

inner regions becomes so hot that it will emit vast amounts of radiation (mainly X-rays), whichmay be detected by telescopes.

This process of accretion is one of the most efficient energy-producing processes known; up to

40% of the rest mass of the accreted material can be emitted in radiation. (In nuclear fusion onlyabout 0.7% of the rest mass will be emitted as energy.)



/91

20.4. No Hair

Black holes have no hair : core collapse loses all properties of the star. The black hole is

characterized completely by:

1. mass

2. angular momentum. Black holes turn very rapidly, typically in a few ms

3. electric charges

20.5. Cosmic censorship conjecture

Cosmic censorship conjecture: singularities of the black hole are hidden inside horizon. Nothing

that happens within the black hole matters for the outside, because it has no effect.

We may observe other singularities outside of a black hole. They are called naked singularities.

20.6. Hawking radiation

Quantum effect near the horizon leads to radiation with energy loss. Because T of a black hole is

inversely proportional to its mass, the bigger the black hole, the colder it is, and therefore the

more difficult it would be to notice it. Therefore, mass of a black hole decreases with time,temperature increases, and at the end of its life, it evaporates through a cosmic explosion.

T in Kelvin, h is the reduced Planck constant, M the mass of

the black hole.

Expected lifetime of a 5 Msun black hole would be 1062

years. Maybe we could see it with 2kg“microscopic” black holes.

20.7. Wormholes

An Einstein-Rosen Bridge (or wormhole) is a hypothetical topological

feature of spacetime that would be, fundamentally, a "shortcut" through

spacetime.

There is no observational evidence for wormholes, but on a theoretical

level there are valid solutions to the equations of the theory of generalrelativity which contain wormholes.

The first type of wormhole solution discovered was the Schwarzschild

wormhole which would be present in the Schwarzschild metricdescribing an eternal black hole, but it was found that this

type of wormhole would collapse too quickly for anything to

cross from one end to the other. Wormholes which could

actually be crossed in both directions, known as traversable

wormholes, would only be possible if exotic matter with

negative energy density could be used to stabilize them.



/91

20.8. Example: compute wavelength of X-ray emission of the accretion

disk surrounding black hole

• X-ray spectrum shows peak at + observed = 2.068e-10m.

• Computations show disk is orbiting black hole at distance R=3 R s

a) Correcting for gravitational effect, we find wavelength peak at:

=1.69e-10m

b) Computing speed of disk using =0.41c we can correct for Doppler effect due

to high speed:

where + 0 is previously corrected + . 5 = 1.54e-10m.



/91

21. Galaxies

21.1. The Milky way

• Diameter: 100-120 l years (31-37 kpc)

• Thickness: 1k lyears (0.31 kpc)

• Sun is about 8kpc from center

• Thin disk: 6. 1010 Msun, young (8 Gy) stars

25x.35kpc

• Thick disk: 3.109 Msun, older (10-11Gy)

stars, 25x1kpc

• Central Buldge: 1010 Msun, stars of all ages,

5x2kpc

• Halo: 3.109 Msun, oldest (11-13 Gy) stars

21.2. Tracking matter

The matter we can see depends on the band frequency:

21.3. The Milky way disk structure

2 main arms.



/91

21.4. The Milky Buldge and Core

• Dense molecular cloud, but no ongoing star formation

• The motion of material around the center indicates that Sagittarius A* harbors a massive,

compact object. This concentration of mass is best explained as a supermassive black

hole with an estimated mass ~ 4 million times the mass of the Sun. Observations indicate

that there are supermassive black holes located near the center of most normal galaxies.

21.5. The Milky Halo

• The Galactic disk is surrounded by a spheroidal halo of old stars and globular clusters, of

which 90% lie within 100,000 light-years (30 kpc) of the Galactic Center

• The temperature of this halo was said to be between 1 million and 2.5 million kelvin or a

few hundred times hotter than the surface of the sun, stated by scientists

• On September 24, 2012, a team of five astronomers working with the Chandra X-ray

Observatory, along with data gathered by the XMM-Newton, and Suzaku (satellite)missions, announced that the halo had a mass nearly equivalent to the galaxy itself

• On January 9, 2006, Mario Juri6 and others of Princeton University announced that the

Sloan Digital Sky Survey of the northern sky found a huge and diffuse structure (spreadout across an area around 5,000 times the size of a full moon) within the Milky Way that

does not seem to fit within current models. The collection of stars rises close to

perpendicular to the plane of the spiral arms of the Galaxy. The proposed likely

interpretation is that a dwarf galaxy is merging with the Milky Way.

21.6. Weighting the Milky way

• Sun orbits the Milky way at 220 km/s at 8kpc, with a period of 230 My (so the Sun has

orbited 20-25 times in its life so far)

• Using the Sun’s orbiting period and its distance from the center of the galaxy, we can

estimate the total mass of the galaxy:

• Or knowing the Sun’s speed we can also use: where M(R) is the mass

within the R radius



/91

where C is a constant depending on how vM compares to the velocity

dispersion of the surrounding matter.

• In close encounters, tidal forces break spreading it in stellar steam

• When 2 galaxies meet, the combined galaxy could have 2 black holes

•

The detailed process by which such early galaxy formation occurred is a major openquestion in astronomy. Theories could be divided into two categories: top-down and

bottom-up. In top-down theories (such as the Eggen–Lynden-Bell–Sandage [ELS]

model), protogalaxies form in a large-scale simultaneous collapse lasting about one

hundred million years. In bottom-up theories (such as the Searle-Zinn [SZ] model), smallstructures such as globular clusters form first, and then a number of such bodies accrete

to form a larger galaxy.

21.10. Measuring distance to galaxies: Redshift

• If galaxy is not too far (< 107 pc), we can use Cepheids standard candles

• For further, spiral galaxies, Tully-Fisher relation relates rotation to luminosity & type

• For further, elliptical galaxies, Fundamental plane relation relates luminosity to size &velocity dispersion

• This allows galaxies to be used as standard candles for distance measurement

• Red shift indicates galaxy speed. What Hubble noticed, is that the farther the galaxy, the

faster it goes away from us : v=H0.D Hubble law H0=100.h km/s/Mpc

h=0.71 hubble constant

This law gives the speed of a galaxy (in km/s), 1 megaparsec away.

Redshift z:

From this is follows that we can deduce the galaxy speed straight from the Doppler shift:

for small Doppler shifts

21.11. Cosmic expansion

• In fact, Hubble constant varies with time: the rate of expansion is not constant intime. For unbound objects:

D(t) = D(t0). [1 + H0(t-t0)] where D(t) is the distance to a galaxy, measured at time t.

t0 is now

• Earth, Sun etc are bounded, therefore they are not expanding vs. each other

• From this, we can deduce the age of the Universe: 1 + H0.(t-t0) = 0, which gives t = t0

- H0-1 ! concerting everything to same units: H0

-1 = 13.8Gy ago

• With the redshift we can compute the distance to the galaxy:

• With the redshift we can also compute how far ago the light that we observe from the

galaxy has been emitted:



/91

• This can be expressed also as: for z <<1 which means that the emitted

wavelength expands with the universe. The Redshift is a cosmological shift, not a

Doppler shift.

21.12. Recap on formulasFrom a star orbiting a galaxy:

• From a star’s orbiting period, its mass and its distance from the center of the galaxy, we

can estimate the total mass of the galaxy:

• From the Star’s speed we can estimate the total mass of the galaxy:

where M(R) is the mass within the R radius

Using the galaxy’s redshift:

• From the distance to a galaxy, we can find its speed: V = H0.D (Hubble law)

H0=100.h km/s/Mpc, h=0.71

• Based on shift of wavelength emitted by galaxy, we compute the redshift:

• We can compute galaxy’s speed: % z for small Doppler shifts

• We can compute galaxy’s distance from us: or for z <<1, i.e. for

speed low enough, e.g. D << c/H0.

• We can compute the time when the light that we observe from the galaxy has been

emitted:

• From the time of light journey, we can also estimate the distance to the galaxy

We could also use which gives where

since + 0 is + observed now.

The above relations ignore the relativist corrections needed at high speed, and the history ofthe universe: H0 is now, we need to integrate trough all previous H.

In deducting the age of the universe we assumed H constant, which is not the case:

the gravity effect will bring back all galaxies together, therefore the expansion will slow and

become negative. Or, if speeds are higher than escape velocity, it will increase forever.



/91

21.13. Galaxy clusters

• Milky way is part of a cluster, or local group.

• Estimated mass of 4.1022 Msun , most of it is dark matter

• Virgo cluster has 250 large galaxies and over 2000 smaller ones. 68% spirals and 19%

ellipticals.

• Intracluster medium of hot 106 gas contains 8 times more mass than galaxies

• Intergalactic stars account for 10% mass• Gravitational lensing by clusters can be used to find mass

distribution of lens:

• Most of the dark matter is diffuse

• Dark matter interacts weakly so follows galaxies

• Clusters are organized in Superclusters. Our supercenter is

centered on Virgo galaxy

• Superclusters are grouped into metastructures

• Correlation data shows after 100Mpcs universe is homogeneous: there is no larger

structure



/91

22. Cosmology

22.1. The cosmological principle

The cosmological principle is the assertion that the universe, at the largest scales, is

homogeneous and isotropic.Homogeneous: Every position is equivalent. There is no center or edge.

Isotropic: Every direction is equivalent.

The cosmological principle is true if we go very far away (100 Mpc), where there is no more

perturbations due to stars or galaxies.

In General Relativity, Homogeneous isotropic universe

means that we can find a coordinates in which curvature

is constant. This does not mean that the universe is flat. In

fact, there are three solutions.

The sum of the angles of a triangle is 180° only if the

universe if flat. If it’s close, it’s more that 180°. If it’s

open, it’s less. But for small distances, it looks 180°.

22.2. Robertson-Walker model

The Robertson–Walker model is an exact solution of Einstein's field equations of generalrelativity; it describes a homogeneous, isotropic expanding or contracting universe that may be

simply connected or multiply connected. This model is sometimes called the Standard Model of

modern cosmology. This model is compatible with the Cosmological redshift.

If we assume that the universe if expanding, distance will grow between now (t0) and the future,

because the universe sphere will grow. The scale factor a(t) gives the increase in distance

between comoving observers: D(t) = a(t) . D0 where we choose a(t0)=1.

Light moves along geodesics of space (e.g. straightest line)

• D(t) = D(t0). [1 + H0.(t-t0)] for H0|t-t0| <<1

• We can estimate acceleration at time t* based on acceleration at time t

From now on, we will assume the universe is a RW model.

22.3. Angular size distance (k=0)

Remember that the redshift indicates by how much the distance from the object has expanded,since the light has been emitted, since the light beam has been continuously expanded during its

journey. Therefore, to obtain the real distance from a distance we observe, we need to correct it by dividing by (1+z).

We define the angular size distance (or also angular diameter distance) as:



/91

where D0 is the distance observed to the object (when light was emitted), and DA is the real

distance to the object (when light was emitted) factoring the redshift in. This is NOT the distance

to the object now.

Objects appear bigger that they really were.

22.4. Luminosity distance (k=0)Observing the luminosity of a star, light that we observe has lost energy because of the redshift

(!(1+z) factor), and there are less photons by unit of time (!(1+z) factor). Therefore, the real

brightness is where:

Luminosity distance and the real brightness,

factoring the redshift in.

22.5. Correcting the temperature for redshift

Using Wien law ( # max . T= b), we deduce that temperatures observed get also affected by the

redshift:

Distance used to

compute brightness

Distance observed, or

comoving transversedistance

Distance real

Everything is observed in the past



/91

22.6. Correcting the galaxy speeds for redshift

Redshift also means that clocks observed in the past seem to run slower.

Since v=x/t, galaxies observed in the past seem to speed 4 times slower than they really did.

22.7. Einstein field equations

The Einstein field equations (EFE) may be written in the form:

and

where R 0 is R at t=t0 and k=

where R µ - is the Ricci curvature tensor, R the scalar curvature of space, gµ - the metric tensor, G

is Newton’s gravitational constant, c the speed of light in vacuum, and Tµ - the stress–energy

tensor.

The EFE is a tensor equation relating a set of symmetric 4 x 4 tensors. Each tensor has 10

independent components.

The scalar curvature of space, R, represents the amount by which thevolume of a ball in a curved space deviates from that of the standard ball in

Euclidean space.

The Ricci curvature tensor R µ -, named after Gregorio Ricci-Curbastro,represents the amount by which the volume element 7V in a curved space

deviates from a flat Euclidean space, to a curved space. Indeed in a curved

space, the sum of the angles of a triangle is not 180°.

The metric tensor gµ - captures all the geometric and causal structure of spacetime. For example,

in special relativity, the metric tensor is , which corresponds to

.

The stress-energy tensor Tµ - (or stress-energy-momentum tensor)

describes the density of energy and the flux of energy inspacetime, generalizing the stress tensor of Newtonian physics

(which is a measure of the average force per unit area of a surfacewithin the body on which internal forces act). It is an attribute of

matter, radiation, and non-gravitational force fields.

The stress–energy tensor is the source of the gravitational field in the Einstein field equations of

general relativity, just as mass density is the source of such a field in Newtonian gravity.

When gravity is negligible, Tµ - = 7 Tµ - = 0

The cosmological constant was introduced by Einstein to make the universe static (as he though

it was) – but Hubble showed it was not needed, since the universe is not static. For Einstein:

P=0

k=1 positive curvaturek=0 flat universe

k=-1 negative curvatureCosmological constant



/91

The CMBR has a thermal black body spectrum at a temperature of 2.7 K. The spectral density

peaks in the microwave range of frequencies.

The glow is very nearly uniform in all directions, but the tiny residual variations show the same

pattern as that expected of a fairly uniformly distributed hot gas that has expanded to the current

size of the universe. In particular, the spatial variation in spectral density (the derivative of the

spectral density function with respect to the angle of observation in the sky) contains smallanisotropies, or irregularities, which vary with the size of the region examined. They match whatwould be expected if small thermal variations, generated by quantum fluctuations of matter in a

very tiny space, had expanded to the size of the observable universe we see today.

This is the most distance light that we will ever see!

Angular Power Spectrum of the CMBIn the case of the CMB, the sky is divided up into polar coordinates 1 and 9, and the observed

temperature field can be decomposed into

spherical harmonics, via the following formula:

A 2-dimensional angular power spectrummeasures the power of a particular angular scale.

This is the Angular power spectrum of theCMB: the power of the temperature, as a function

of the angular scale (where the sky is divided upinto 90° angle). A small angular scale means that

this is the temperature’s power that we observe in

front of us, while a larger angular scale (90°)means the temperature's power above us.

1. The angular scale of the first peak determines the curvature of the universe (but not the

topology of the universe). Since the angle is very low, we conclude the universe is nearly

flat (e.g. within a 1° angle).

2. The next peak—ratio of the odd peaks to the even peaks—determines the reduced baryon

density.

3. The third peak can be used to get information about the dark matter density.

22.17. Big Bang NucleosynthesisBig Bang nucleosynthesis (or primordial nucleosynthesis, BBN) refers to the production of

nuclei other than those of H-1 (i.e. the normal, light isotope of hydrogen, whose nuclei consist of

a single proton each) during the early phases of the universe. Primordial nucleosynthesis took

place just a few moments after the Big Bang and is responsible for the formation of a heavier

isotope of hydrogen known as deuterium (H2 or D), the helium isotopes He

3 and He

4, and the

lithium isotopes Li6 and Li7. In addition to these stable nuclei some unstable, or radioactive,

isotopes were also produced: tritium (H3), beryllium (Be

7), and beryllium (Be

8). These unstable

isotopes either decayed or fused with other nuclei to make one of the stable isotopes.

Two important characteristics:

• The corresponding time interval was from a few tenths of a second to up to 103 seconds

• It was widespread, encompassing the entire observable universe.



/91

22.18. LCDM Cosmology

Lambda Cold Dark Matter (LCDM) Cosmology is a parametrization of the Big Bang

cosmological model in which the universe contains a cosmological constant, denoted by

Lambda, and cold dark matter. It is frequently referred to as the standard model of Big Bang

cosmology.

The universe is probably infinite, but the observable universe is not.

Big bang’s curvature was infinite.

3 problems have to be solved:

1. Why is the CMBR so uniform? Different regions of the universe have not "contacted"each other because of the great distances between them, but nevertheless they have the

same temperature and other physical properties. This should not be possible, given thatthe transfer of information (or energy, heat, etc.) can occur, at most, at the speed of light.

2. Why is the universe so close to being flat today?From the Friedmann equations, it appears that 8total is very sensitive: add 1 gram, and the

universe collapses. Remove 1 gram, and the universe will expand forever. So, in order to

reach 8total % 1 now, 8total must have been VERY close to 1 in the early universe.

3. Why are there no magnetic monopoles in the universe today?

The extensions of standard model (GUTs) predict the formation of stable magnetic

monopoles and other defects. But searches discover no monopoles. Where are they?

22.19. Inflation

Cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid

exponential expansion of the early universe by a factor of at least 10 78 in volume, driven by a

negative-pressure vacuum energy density. It lasted from 10336 seconds after the Big Bang to

sometime between 10333 and 10332 seconds.

Following the inflationary period, the universe continued to expand, but at a slower rate.The inflationary hypothesis was originally proposed in 1980 by American physicist Alan Guth,

who named it "inflation".

Inflation theory solves the 3 problems:

1. The CMBR is uniform because all regions were connected

2. while H is constant, therefore since at inflation, the scale factor a

became huge, 8 was forced to 1.

3. Relics have been diluted by inflation

The multiverse (or meta-universe) is the hypothetical set of multiple possible universes

(including the historical universe we consistently experience) that together comprise everything

that exists and can exist: the entirety of space, time, matter, and energy as well as the physical

laws and constants that describe them. Each universe would have its own cosmologicalconstants. They would arise because small regions of the universe could reach different false

vacua of energy, thereby reaching different inflation rates. They would expand less than theremaining universe, leading to disconnected patched and disappearing for our universe.

We are now in exponential inflation, which leads to freezing fluctuations, forming the seeds for

structure formations leading to clusters and galaxies.



/91

22.20. Exercise: compute the distance when the light was emitted,and the distance now, from a galaxy

• From the redshift z measured now, we can compute the scale factor at the time the object

emitted that light:

• From the scale factor, we can compute the time t when the light was emitted, or when the

galaxy is observed. In dust matter-dominated era, t being the time since the beginning of

the era: or ! t in sec since the era’s beginning

• To find how long ago the light was emitted, we need to compute t0-t where t0 is the time

from the beginning of the era to now. Now: a(t0)=1! t0 = 2/(3.H0).

• From the scale factor a and the time t0 when light was emitted, we can compute the

distance NOW to the object. In dust matter-dominated era:

t0 is the time now since the beginning of the era. This is not t0-t.

Notice that a is the scale factor at the time the object emitted that light, i.e. computed

from the redshift. It is not the scale factor today.

• From the redshift and the current distance D0 to the object, we can find the distance to the

object when the light was emitted:

• To compute the temperature of the universe (e.g. CMB) at that time, we can use

where Tobs is the temperature of the CMB observed now (2.726K), and Tem

the temperature of the CMB at the time.

22.21. Exercise: compute the distance angular radius of an object

If we know the radius R of the object, we can evaluate its angular radius using

22.22. Exercise: compute the brightness of an object, knowing itsluminosity

From the Luminosity we can compute the brightness observed, using where

(one factor comes from the loss of energy because of the redshift, a second

factor because there are less photons by unit of time).

22.23. Exercise: compute the observed luminosity period,knowing its real luminosity period (e.g. when light was emitted)

If a star has a luminosity period P of 40 days, to compute the observed luminosity period Pobs, wesimply need to take into account time expansion: Pobs=P.(1+z)



22.24. Plasma and Ionization

A collection of non-aqueous gas-like ions, or more generally a

gas containing a proportion of charged particles, is called a

plasma. Greater than 99.9% of visible matter in the Universe

may be in the form of plasmas. These include our Sun and

other stars and the space between planets, as well as the spacein between stars. Plasmas are often called the fourth state ofmatter because their properties are substantially different from

those of solids, liquids, and gases. Astrophysical plasmas predominantly contain a mixture of electrons and protons

(ionized hydrogen).

Ionization is the process ofconverting an atom or molecule

into an ion by adding or

removing charged particles suchas electrons or ions. In the caseof ionization of a gas, ion pairs

are created consisting of a free

electron and a positive ion.

The Ionization energy of an atom and its temperature are linked through:

at z = 0

At z : 0, where T0 is the temperature now (at z=0), and Tz the temperature of the at

the time of the redshift.

The proportion of H atoms with enough energy to ionize Hydrogen are given by:

introduction to astronomy (coursera)

Documents