Stellar Spectra

• Flux and Luminosity• Brightness of stars• Temperature vs heat• Temperature vs color• Colors/spectra of stars• Classifying stars: H-R diagram• Stefan-Boltzman law, stellar radii• Measuring star masses• Mass-luminosity relation

Flux and luminosity

• Luminosity - A star produces light – the total amount of energy that a star puts out as light each second is called its Luminosity.

• Flux - If we have a light detector (eye, camera, telescope) we can measure the light produced by the star – the total amount of energy intercepted by the detector divided by the area of the detector is called the Flux.

Flux and luminosity

• To find the luminosity, we take a shell which completely encloses the star and measure all the light passing through the shell

• To find the flux, we take our detector at some particular distance from the star and measure the light passing only through the detector. How bright a star looks to us is determined by its flux, not its luminosity. Brightness = Flux.

Flux and luminosity

• Flux decreases as we get farther from the star – like 1/distance2

24 D

LF

Brightness of stars

• The brightness of a star is a measure of its flux.• Ptolemy (150 A.D.) grouped stars into 6

`magnitude’ groups according to how bright they looked to his eye.

• Herschel (1800s) first measured the brightness of stars quantitatively and matched his measurements onto Ptolemy’s magnitude groups and assigned a number for the magnitude of each star.

Brightness of stars

• In Herschel’s system, if a star is 1/100 as bright as another then the dimmer star has a magnitude 5 higher than the brighter one.

• Note that dimmer objects have higher magnitudes

Consider two stars, 1 and 2, with apparent magnitudes m1 and m2 and fluxes F1 and F2. The relation between apparent magnitude and flux is:

2

11021 log5.2

F

Fmm

Apparent Magnitude

5.2/

2

1 1210 mm

F

F

For m2 - m1 = 5, F1/F2 = 100.

Flux, luminosity, and magnitude

1

210

1

21012

2

22

21

11012

2

11012

2

log5log5.2

4

4log5.2

log5.2

4

D

D

L

Lmm

L

D

D

Lmm

F

Fmm

D

LF

Distance-Luminosity relation:Which star appears brighter to the

observer?

d

10d

L10L

Star 1

Star 2

Flux and luminosity

10D

D

1

2 10L

L

1

2

1.010

110

4

4

22

2

1

1

2

1

21

22

2

1

2

D

D

L

L

L

D

D

L

F

F

Star 2 is dimmer and has a higher magnitude.

Flux and luminosity

10D

D

1

2 10L

L

1

2

5.255.2

10log510log5.2

log5log5.2

12

101012

1

210

1

21012

mm

mm

D

D

L

Lmm

Star 2 is dimmer and has a higher magnitude.

Absolute magnitude• The magnitude of a star gives it brightness

or flux when observed from Earth.

• To talk about the properties of star, independent of how far they happen to be from Earth, we use “absolute magnitude”.

• Absolute magnitude is the magnitude that a star would have viewed from a distance of 10 parsecs.

• Absolute magnitude is directly related to the luminosity of the star.

Absolute magnitude, M, is defined as

Absolute Magnitude

5log5 10 DmM

where D is the distance to the star measured in parsecs.For a star at D = 10 parsecs, 5log10 = 5, so M = m.

Absolute Magnitude and Luminosity

The absolute magnitude of the Sun is M = 4.83.The luminosity of the Sun is L

L

LM

LL M

10

5.2/83.4

log5.283.4

10

Note the M includes only light in the visible band, so this is accurate only for stars with the same spectrum as the Sun.

Absolute Bolometric Magnitude and Luminosity

The bolometric magnitude includes radiation at all wavelengths.The absolute bolometric magnitude of the Sun is Mbol = +4.74.

L

LM bol 10log5.274.4

Is Sirius brighter or fainter than Spica:

(a) as observed from Earth [apparent magnitude] (b) Intrinsically [luminosity]?

Sun

Little Dipper

(Ursa Minor)

Guide to naked-eye magnitudes

Which star would have the highest magnitude?

1. Star A - 10 pc away, 1 solar luminosity

2. Star B - 30 pc away, 3 solar luminosities

3. Star C - 5 pc away, 0.5 solar luminosities

4. Charlize Theron

Temperature

• Temperature is proportional to the average kinetic energy per molecule

lower T higher T

kTmvK2

3

2

1 2 k = Boltzmann constant = 1.3810-23 J/K = 8.6210-5 eV/K

Temperature vs. Heat• Temperature is proportional

to the average kinetic energy per molecule

• Heat (thermal energy) is proportional to the total kinetic energy in box

lower T higher T

same T

less heat more heat

Wien’s law

• Cooler objects produce radiation which peaks at lower energies = longer wavelengths = redder colors.

• Hotter objects produce radiation which peaks at higher energies = shorter wavelengths = bluer colors.

• Wavelength of peak radiation:Wien Law max = 2.9 x 106 / T(K) [nm]

A object’s color depends on its surface temperature

• Wavelength of peak radiation:Wien Law max = 2.9 x 106 / T(K) [nm]

What can we learn from a star’s color?

The color indicates the temperature of the surface of the star.

U B V

Observationally, we measure colors by comparing the brightness of the star in two

(or more) wavelength bands.

This is the same way your eye determines color, but the bands are different.

Use UVRI filters to determine apparent

magnitude at each color

Stars are assigned a `spectral type’ based on their spectra

• The spectral classification essentially sorts stars according to their surface temperature.

• The spectral classification can also use spectral lines.

Spectral type• Sequence is: O B A F G K M• O type is hottest (~25,000K), M type is coolest

(~2500K)• Star Colors: O blue to M red• Sequence subdivided by attaching one numerical

digit, for example: F0, F1, F2, F3 … F9 where F1 is hotter than F3 . Sequence is O … O9, B0, B1, …, B9, A0, A1, … A9, F0, …

• Useful mnemonics to remember OBAFGKM:– Our Best Astronomers Feel Good Knowing More– Oh Boy, An F Grade Kills Me– (Traditional) Oh, Be a Fine Girl (or Guy), Kiss Me

Very faint

The spectrum of a star is primarily determined by

1. The temperature of the star’s surface

2. The star’s distance from Earth

3. The density of the star’s core

4. The luminosity of the star

Classifying stars

• We now have two properties of stars that we can measure:– Luminosity– Color/surface temperature

• Using these two characteristics has proved extraordinarily effective in understanding the properties of stars – the Hertzsprung-Russell (HR) diagram

HR diagram

HR diagram• Originally, the HR diagram was made

by plotting absolute magnitude versus spectral type

• But, it’s better to think of the HR diagram in terms of physical quantities: luminosity and surface temperature

If we plot lots of stars on the HR diagram, they fall into groups

These groups indicate types of stars, or stages in the evolution of stars

Stephan-Boltzmann Law: an opaque object at a given temperature will radiate, per unit surface area, at a rate proportional to the surface temperature to the fourth power :

P/m2 = T4

= Stephan-Boltzman constant = 5.6710-8 W/m2 ·K4

T = surface temperatureP/m2 = power radiated per square meter

Luminosity of a ‘Black Body’ Radiator

For the spherical object, the total power radiated = the total luminosity is:

L = 4R2T4

T = temperature = Stephan-Boltzman constant = 5.6710-8 W/m2 ·K4

R = radius

Luminosity of a ‘Black Body’ Radiator

Suppose the radius of the Sun increased by a factor of 4 but the rate of power generated by fusion remained the same, how would the surface temperature of the Sun change?

Luminosity of a ‘Black Body’ Radiator

42

22

41

21 44 TRTRL

11

2

1

1

2

1

2

12 2

1

4

1TTT

R

RT

• If we know luminosity and temperature, then we can find the radius:

L = 4R2T4

• Small stars will have low luminosities unless they are very hot.

• Stars with low surface temperatures must be very large in order to have large luminosities.

Stars come in a variety of sizes

Sizes of Stars on an HR Diagram

• We can calculate R from L and T.

• Main sequence stars are found in a band from the upper left to the lower right.

• Giant and supergiant stars are found in the upper right corner.

• Tiny white dwarf stars are found in the lower left corner of the HR diagram.

Hertzsprung-Russell (H-R) diagram

• Main sequence stars– Stable stars found on a

line from the upper left to the lower right.

– Hotter is brighter– Cooler is dimmer

• Red giant stars– Upper right hand corner

(big, bright, and cool)• White dwarf stars

– Lower left hand corner (small, dim, and hot)

Luminosity classes

• Class Ia,b : Supergiant• Class II: Bright giant• Class III: Giant• Class IV: Sub-giant• Class V: Dwarf

The Sun is a G2 V star

‘Spectroscopic Parallax’Measuring a star’s distance by

inferring its absolute magnitude (M) from the HR diagram

1. If a star is on the main-sequence, there is a definite relationship between spectral type and absolute magnitude. Therefore, one can determine absolute magnitude by observing the spectral type M.

2. Observe the apparent magnitude m.

3. With m and M, calculate distance

Take spectrum of star, find it is F2V, absolute magnitude is then M = +4.0.

Observe star brightness, find apparent magnitude m = 9.5.

Calculate distance:

pc 126pc 10 5log5 5/)55.5(10 DDMm

Masses of stars

• Spectral lines also allow us to measure the velocities of stars via the Doppler shift that we discussed in searching for extra-solar planets. Doppler shift measurements are usually done on spectral lines.

• Essentially all of the mass measurements that we have for stars are for stars in binary systems – two stars orbiting each other.

• The mass of the stars can be measured from their velocities and the distance between the stars.

• Double star – a pair of stars located at nearly the same position in the night sky.– Optical double stars – stars that appear close together,

but are not physically conected.– Binary stars, or binaries – stars that are gravitationally

bound and orbit one another.

• Visual binaries – true binaries that can be observed as 2 distinct stars

• Spectroscopic binaries – binaries that can only be detected by seeing two sets of

lines in their spectra– They appear as one star in telescopes (so close

together)

• Eclipsing binaries – binaries that cross one in front of the other.

Binary star systems Classifications

Visual Binary Star Krüger 60 (upper left hand corner)

About half of the stars visible in the night sky are part of multiple-star systems.

Kepler’s 3rd Law applied to Binary Stars

Where:

• G is gravitational constant

• G = 6.67·10-11 m3/kg-s2 in SI units

• m1, m2 are masses (kg)

• P is binary period (sec)

• A is semi-major axis (m)

23

21

2

)(

4Pa

mmG

Simplified form of Kepler’s 3rd law using convenient units

Where M in solar masses

a in AU

P in Earth years

Example: a = 0.05 AU, P = 1 day = 1/365 yr, M1 + M2 = 16.6 Msun

2

3

21 P

aMM

0.008"

Mizer-Alcor : A double-double-double system!

10 arcmin

14"

Mizar A+B

Alcor

Mizar A

Mizar B

Mizar A

(Binary, P = 20.5 days)

Note: Mizar B is also a binary

with period of 6 months!

Historical Notes on Mizar-Alcor discoveries• Romans (c. 200BC): Used Mizar-Alcor (11 arcmin

separation) as test of eyesight for soldiers

• Benedi Castelli (c. 1613, student and friend of Galileo) discovers Mizar is a double star (separation 15")

– "It's one of the beautiful things in the sky and I don't believe that in our pursuit one could desire better", remarked Castelli in letter to Galileo

– Both Galileo and Castelli were interested in ‘optical doubles’ to prove the heliocentric view of solar system (nearer star would move w.r.t more distant star annually)

• Johann Liebknecht (1722) announced that the 8thmag star SW of Mizar was a new planet! (Incorrect observation of motion)

– He named it Sidus Ludoviciana (Ludwig’s Star) in honor of his local monarch King Ludwig.

• 1887: Pickering at Harvard announces Mizar A is a spectroscopic binary, 20.5 day period

• 1996: NPOI directly images the Mizar A binary (separation 0.008 arcsec)

Mizar A+B

Alcor

Sidus Ludoviciana

Mizar A: A Spectroscopic Binary• 1887 Spectroscopy of Mizar A shows periodic

doubling of spectral lines, with 20.5 day period

Note: Asymmetric light curves indicated ellipical orbits

Mizar observations using the NPOI

(Naval Prototype Optical Interferometer, near Flagstaff Arizona)

Determining masses of Mizar-A binary stars from observations of period, angular separation, distance

1. Distance (from parallax) d = 24 pc (88ly)

3. Physical separation D = θ·d=0.19 AU

2. Max. angular separation (NPOI meas.) Θ = 0.008"

4. Sum of masses (Kepler’s 3rd law)

5. Orbit shows a1 ~ a2 (NPOI meas.) so:

2.2056.0

19.02

3

2

3

21 P

aMM

MMM

aMaM

1.121

2211

Spectroscopy makes it possible to study binary systems in which the two

stars are very close together.

1. Determine semi-major axis using observed velocity (V), period (P)

a1 a2

a = a1 + a2

Determining component masses of eclipsing binaries using velocity curves

2. Determine sum of masses using Kepler’s 3rd law

3. Determine mass ratio using a1, a2

4. Use sum, ratio to determine component masses

21

22

11

2

2

aaaP

av

P

av

2

3

21 P

aMM

22112211 or vMvMaMaM

We have been assuming that we see the binary system face on when imaging the orbit and edge-on when measuring the velocity.

In general, the orbit is tilted relative to our line of sight. The tilt, or inclination i, will affect the observed orbit trajectory and the observed velocities. In general, one needs both the trajectory and the velocity to completely determine the orbit or some independent means of determining the inclination.

Tilt of Binary Orbits

Light curves of eclipsing binaries provide detailed information about the two stars.

Light curves of eclipsing binaries provide detailed information about the two stars.

Eclipsing Binary EQ Tau

Light curve from Astronomical

Laboratory Course Fall 2003

Java-animation of binary stars

Eclipse of an Exo-planet (HD209458)

“Vogt-Russell” theorem for spheres of water

• Spheres of water have several properties: mass, volume, radius, surface area …

• We can make a “Vogt-Russell” theorem for balls of water that says that all of the other properties of a ball of water are determined by just the mass and even write down equations, i.e. volume = mass/(density of water).

• The basic idea is that there is only one way to make a sphere of water with a given mass.

“Vogt-Russell” theorem

• The idea of the “Vogt-Russell” theorem for stars is that there is only one way to make a star with a given mass and chemical composition – if we start with a just formed protostar of a given mass and chemical composition, we can calculate how that star will evolve over its entire life.

• This is extremely useful because it greatly simplifies the study of stars and is the basic reason why the HR diagram is useful.

Mass - Luminosity relation for main-sequence stars

Mass in units of Sun’s mass

Mass-Luminosity relation on the main sequence

5.3

M

M

L

L

• The lifetime of a star (on the main sequence) is longer if more fuel is available and shorter if that fuel is burned more rapidly

• The available fuel is (roughly) proportional to the mass of the star

• From the previous, we known that luminosity is much higher for higher masses

• We conclude that higher mass star live shorter lives

Mass-Lifetime relation

5.25.3

1

MM

M

L

Mt

A ten solar mass star has about ten times the sun's supply of nuclear energy. Its luminosity is 3000 times that of the sun. How does the lifetime of the star compare with that of the sun?

1. 10 times as long

2. the same

3. 1/300 as long

4. 1/3000 as long

300

1

3000

10

L

Mt

Mass-Lifetime relation

Mass/mass of Sun Lifetime (years)

60 400,000

10 30,000,000

3 600,000,000

1 10,000,000,000

0.3 200,000,000,000

0.1 3,000,000,000,000

Stellar properties on main sequence

• Other properties of stars can be calculated such as radius (we already did this).

• The mass of a star also affects its internal structure

(solar masses)

Evolution of stars

• We have been focusing on the properties of stars on the main sequence, but the chemical composition of stars change with time as the star burns hydrogen into helium.

• This causes the other properties to change with time and we can track these changes via motion of the star in the HR diagram.

HW diagram for people

• The Height-Weight diagram was for one person who we followed over their entire life.

• How could we study the height-weight evolution of people if we had to acquire all of the data from people living right now (no questions about the past)?

• We could fill in a single HW diagram using lots of different people. We should see a similar path.• We can also estimate how long people spend on particular parts of the path by how many people we find on each part of the path.

HR diagram