Chapter 10

Measuring the Stars

Read material in Chapter 10

Some of the topics included in this chapter

• Stellar parallax

• Distance to the stars

• Stellar motion

• Luminosity and apparent brightness of stars

• The magnitude scale

• Stellar temperatures

• Stellar spectra

• Spectral classification

• Stellar sizes

• The Herzsprung-Russell (HR)diagram

• The main sequence

• Spectroscopic parallax

• Extending the cosmic distance

• Luminosity class

• Stellar masses

Parallax is the apparent shift of an object relative to some distant

background as the observer’s point of view changes

It is the only direct way to measure distances to stars

It makes use Earth’s orbit as baseline

Parallactic angle = 1/2 angular shift

A new unit of distance: Parsec

By definition, parsec (pc) is the distance from the Sun

to a star that has a parallax of 1” (1 arc second)

Parallax Formula:

Distance (in pc) = 1/parallax (in arcsec)

One parsec = 206,265 AU or ~3.3 light-years

As the distance increases to a star, the parallax decreases….

Examples using the parallax formula:

If the measured parallax is 1 arcsec, then the distance of the star is 1 pc. (Distance 1 pc = 1/1 arcsec)

If the measured parallax is 0.5 arcsec, then the distance of the star is 2 pc. (Distance 2 pc = 1/0.5 arcsec) Note: 1 parsec = 3.26 light-years.

The Solar Neighborhood

Let’s get to know

our neighborhood:

A plot of the 30

closest stars within

4 parsecs (~ 13 ly)

from the Sun.

The gridlines are

distances in the

galactic plane (the

plane of the disc of

the Milky Way)

More examples using the parallax formula

The nearest star Proxima Centauri (the faintest star of the triple star

system Alpha Centauri) has a parallax of 0.76 arcsec.

• Therefore, distance = 1 / 0.76 = 1.32 pc (4.29 ly)

• The next nearest star is Barnard’s star, with a parallax of 0.55”

• Therefore, d = 1 / 0.55 = 1.82 pc (5.93 ly)

The Nearest Stars

• From the ground, we can measure parallactic angles of ~1/30 (0.03”) arcsec, corresponding to distances out to ~30 pc (96 ly).

• There are several thousand stars within that distance from the Sun.

• From space (Hipparcos satellite), parallax’s can be measured down to

about 5/1000 arcsec, which corresponds to 200 pc (~660 ly).

• There are several million stars within that distance.

Using the stellar parallax, the distance to these stars can be determined

directly

Stellar Proper Motion

• Parallax is an apparent motion of stars due to Earth

orbiting the Sun.

• But stars do have real space motions.

• Space motion has two components:

1) “line-of-sight” or radial motion (measured through Doppler

shift of emission/absorption lines)

2) “transverse” motion (perpendicular to the line of sight)

observer star

radial motion

transverse space motion

How to determine the two component of the

space motion of a star?

• Use the Doppler shift to determine the radial component. Observe the

shift in wavelengths of the emission or absorption lines. Then apply

the formula of Doppler shift to determine the radial velocity

• Use the proper motion and the distance to determine the transverse

component.

First we need to measure the proper motion. Proper motion is

measured in arc seconds/year. Then we need to know the distance to

the star using parallax so we can determine the transverse component

• Finally use trigonometry to calculate the transverse velocity

• This method works for stars that are nearby so we can measure the

proper motion.

• The total velocity can be calculated using the Pythagorean theorem:

____________________________________

Total velocity = √ [(Radial velocity)² + (Transverse velocity)²]

Stellar Proper Motion: Barnard’s Star

• Two pictures, taken 22 years apart ( Taken at the same time of the year

so it doesn’t show parallax!). Barnard’s star is a red dwarf of magnitude +9.5,

invisible to the naked eye (limit of naked eye is +6)

• Barnard’s star has a proper motion of 10.3 arcsec/year (it is the

star with the largest proper motion)

• Given d = 1.8 pc, this proper motion corresponds to a

“transverse” velocity of ~90 km/s !

Question: What does the proper motion depend on? Answer 1: Space velocity

Answer 2: Distance

Some important definitions and

concepts

• Luminosity is the amount of radiation leaving a star per unit time.

• Luminosity is an intrinsic property of a star.

• It is also referred as the star absolute brightness.

It doesn’t depend on the distance or motion of the observer respect to

the star.

• Apparent brightness or Flux.

When we observe a star we see its apparent brightness, not its

luminosity. The apparent brightness (or flux) is the amount of light

striking the unit area of some light sensitive device such as the human

eye or a CCD. It depends on the distance to the star.

Apparent Brightness and the Inverse Square Law: Proportional to 1/d2

• Light “spreads out” like the

distance squared.

• Through a sphere twice as large,

the light energy is spread out over

four times the area.

(area of sphere = 4d2)

The apparent brightness or Flux

decreases with distance, it is

inversely proportional to the

square of the distance.

It can be determined by:

Luminosity

4d2 Flux =

To know a star’s luminosity we must measure its apparent

brightness (or flux) and know its distance. Then,

Luminosity = Flux *4d2

Luminosity and Apparent Brightness

Two stars A and B of

different luminosity

can appear equality

bright to an observer if

the brightest star B is

more distant than the

fainter star A

The Magnitude Scale 2nd century BC, Hipparchus ranked all visible

stars

He assigned to the brightest star a magnitude

1, and to the faintest a magnitude 6.

Later, astronomer found out that a difference

of 5 magnitudes from 1 (brightest) to 6

(faintest) correspond to a change in brightness

of 100

To our eyes, a change of one magnitude = a

factor of 2.512 in flux or brightness.

The magnitude scale is logarithmic.

Each magnitude corresponds to a factor of

1001/5 2.5

5 magnitudes = factor 100 in brightness.

The apparent magnitude scale was later

extended to negative values for brighter

objects and to larger positive values for

fainter objects

Brightest

Faintest

Equivalence between magnitude and brightness Magnitude Brightness

-1

2.512

0

2.512

1

2.512

2

2.512

3

2.512

4

2.512

5

2.512

6

The change of brightness between magnitude 1 and 6 is 2.512^5 = 100

In general, the difference in brightness between two magnitudes is:

Difference in brightness = 2.512 ^n, where n is the difference in magnitude

Example: What is the difference in brightness between magnitude -1 and +1?

Answer: n=2, difference in brightness = 2.512² = 2.512 x 2.512 = 6.31

Absolute Magnitude is the apparent magnitude of a star as

measured from a distance of 10 pc (33 ly).

Sun’s absolute

magnitude = +4.8

It is the magnitude of

the Sun if it is placed at

a distance of 10 pc.

Just slightly brighter

than the faintest stars

visible to the naked eye

(magnitude = +6) in the

sky.

Enhanced color picture of the sky

Notice the color differences among the stars

Stellar Temperature: Spectra

• The spectra shows 7 stars with

same chemical composition but

different temperatures.

• Different spectra result from

different temperatures.

Example: Hydrogen absorption lines

are relatively weak in the hottest star

because it is mostly ionized.

Conversely, hotter temperatures are

needed to excite and ionize Helium so

these lines are strongest in the hottest

star.

Molecular absorption lines (TiO) are

present in low temperature stars. The

low temperatures allow formation of

molecules Ti Titanium, TiO titanium oxide

Spectral Classification:

Annie Jump Cannon

The stars were classified by the Hydrogen line strength,

and started as A, B, C, D, …

But after a while they realized that there is a sequence in

temperature so they rearranged the letters (some letters were

drop from the classification) so that it reflect a sequence in

temperature. It became:

O, B, A, F, G, K, M, (L)

A temperature sequence! Cannon’s spectral

classification system was officially adopted in 1910.

A classification of

stars was started by

the “Pickering’s

women”, a group of

women hired by the

director of the Harvard

College observatory,

including Annie

Cannon

Spectral Classification

A mnemonic to remember the correct order:

“Oh Be A Fine Girl/Guy Kiss Me”

Each letter is divided in 10 smaller subdivisions from 0 to 9. The lower the number, the hotter the

star. Example, G0 (hotter) to G9 (cooler). The Sun is classified as a G2 star, the surface

temperature is 5800 K

Strengths of Lines at Each Spectral Type

Stellar Radii

• Almost all stars are so distant that the image of their discs look so

small. Their images appear only as an unresolved point of light

even in the largest telescopes. Actually the image shows the Airy

disk produced by the star. • A small number of stars are big, bright and close enough to

determine their sizes directly through geometry.

•Knowing the angular diameter and the distance to the star, it is

possible to use geometry to calculate its size.

Diameter/2π x distance = Angular diameter/360

Stellar Radii • One example in which it is possible to use geometry to determine the radius

is the star Betelgeuse in the Orion constellation

• The star is a red giant located about 640 ly from Earth

• Betelgeuse size is about 600 time larger than the Sun

• Its photosphere exceed the size of the orbit of Mars

• Using the Hubble telescope it is possible to resolve its atmosphere and

measure its diameter directly

• The measured angular size is about 0.043-0.056 arc seconds

An indirect way to determine the stars radii

• Most of the stars are too distant or too small to allow the

direct determination of their size.

• But we can use the radiation laws to make an indirect

determination of their size.

• According to Stefan law, the luminosity of a star is

proportional to the fourth power of the surface temperature

(T4 )

• The luminosity also depend on its surface area. Larger

bodies at the same temperature radiate more energy.

• Luminosity Surface area * T4

Lstar= (Rstar/Rsun)2 * (Tstar/Tsun)

4 * Lsun

Stellar Radii: An indirect way to measure the radius (Read 10-2 More Precisely, “Estimating Stellar Radii’)

Stefan’s Law F = T4

Luminosity (L) is the Flux (F) multiplied

by the entire spherical surface (A)

L = A * F

Area of sphere A = 4R2

Expressing in solar units (dividing

by the solar L, R and T), the

constants disappear:

L = 4R2T4

L = 4R2 F

Flux (F) is the energy radiated per unit area

by a black body at the temperature T

(R is the radius of the star)

Substituting A in the equation of L

Substituting F in the equation of L

The relationship between Luminosity, Radius, and Temperature

provides a means to evaluate these properties relative to the Solar

values.

T

T

R

R

L

TRTR

L

sunsunsun

sunsunsun

L

L

42

42

42

4

4

For example, a star has 10 times the Sun’s radius but is half as hot. (Since this is relative to the Sun, we will consider that the radius of the Sun is 1

and the temperature of the Sun is 1)

How much is the luminosity respect to the Sun?

25.616

100

2

1

1

1042

Lsun

L

Determining radii using radiation laws

The equation L = 4R2T4 can be expressed in solar units as:

• L(in solar luminosities) = R2 (in solar radius) * T4 (in solar surface

temperature)

• If we need to calculate the radius, we can rearrange the equation :

R2 (in solar radius) = L(in solar luminosities) / T4 (in solar surface

temperature)

Here we need to know the luminosity L and T. To determine L, we need

to know the Flux and the distance d. To get T, we need to get the

spectrum of the star.

Luminosity = Flux *4d2

Understanding Stefan’s Law: Radius dependence

Lstar= (Rstar/Rsun)2 * (Tstar/Tsun)

4 * Lsun

If we receive 100 photons from the Sun, we should receive 400 photons from a star twice the

diameter of the Sun. The star will look four times brighter than the Sun

Let’s consider a star that has a radius twice the radius of the Sun.

What will be the luminosity of that star? (We assume that the two stars have the same temperature)

Understanding Stefan’s Law: Temperature dependence

Lstar= (Rstar/Rsun)2 * (Tstar/Tsun)

4 * Lsun

The luminosity of a star that has a temperature twice that of the Sun, must be 16 times

larger.

The luminosity of a star with a temperature 1/3 of the Sun, must be 1/81 that of the Sun

The assumption here is that these stars have the same radius

Let’s consider a star with a temperature twice that of the Sun and

another star with a temperature one third of the Sun

Hertzsprung-Russell (HR) Diagram

The Main Sequence (MS)is the

diagonal band of stars in the HR

diagram

Stars reside in the main sequence

during the period in which the core

burns H

Most stars (like the Sun) lie on the

main sequence. The Sun will spend

most of its life in the main sequence (It

has been in the MS for about 5 billion

years)

The HR diagram is a plot of

star Luminosity versus

Temperature (or spectral

class)

It also give information about:

•Radius

•Mass

•Lifetime

•Stage of Evolution

Main sequence

From Stefan’s law…...

Let’s use the equation and

the HR diagram to learn

more about L, R and T

More luminous stars at

the same T must be bigger!

Cooler stars at the same L

must be bigger!

L = 4R2 T4

The HR diagram to the

right has L and T on

the axes. But we can

plot R (The other

parameter in the

equation) also which

will appear as straight

lines crossing the

diagram

The HR Diagram: 100 Brightest Stars

• Most luminous stars,

because they are so rare, lie

beyond 5 pc.

• If we know the luminosity,

we can determine distance

from their Flux (brightness).

Luminosity

4d2 Flux =

The technique to

determine distances to

stars using the radiation

laws and HR diagram is

called: Spectroscopic

“Parallax”

The HR Diagram: Spectroscopic “Parallax”

Main Sequence

1) We measure the Flux or apparent

brightness of a star

Apparent brightness is the rate at which

energy from the star reaches a detector

2) From the spectrum of a star, we

can determine its temperature or the

spectral type.

3) Then using the HR diagram we can

determine its luminosity assuming

it is located in the Main

Sequence

4) Use inverse square law to

determine distance.

An example to illustrate how this

works:

Luminosity

4d2 Flux =

What if the star doesn’t happen to lie on the Main

Sequence - maybe it is a red giant or white dwarf???

We determine the star’s Luminosity Class based on its

spectral line widths:

Spectral lines

get broader

when the

stellar gas is at

higher

densities -

indicates

smaller star.

A Supergiant

star

A Giant star

A Dwarf star (Main Sequence)

Wavelength

The HR Diagram: Luminosity & Spectroscopic Parallax

The HR Diagram: Luminosity Class

Bright Supergiants

Supergiants

Bright Giants

Giants

Sub-giants

Main-Sequence (Dwarfs)

• Isn’t this getting a little circular?

• First we said that we derive Luminosities from measured Fluxes and

Distances?

• Now we’re saying we know the Luminosities and we use them together with

Temperatures to derive Distances……..

Let’s clarify this!

Example of absorption lines for different spectral classes

The lines are wider for dwarf (denser) stars of spectral class V and

narrower for giant stars of spectral class I.

More on Spectroscopic Parallax

The answer:

• Now we made use of additional information obtained from

the spectral analysis.

• The spectral analysis provide information to determine the

temperature of the star or the spectral classification (Using

the spectrum of the star). To do this, we didn’t know or

need the distance

• Next we also made use of the HR diagram. If we know the

temperature for a main sequence star (or the luminosity

class), then we can deduce the luminosity

We get distances to nearby planets from radar

ranging

If we know the distance (and we can measure

the orbital period), we apply Kepler’s 3rd law to

obtain the distance Earth-Sun (AU)

That sets the scale for the whole solar system

(1 AU). It allows us to get a value for the AU in

km (1AU = 150,000,000 km)

Knowing the value of the AU in km, we use

the stellar parallax, to find distances to “nearby”

stars.

Use these nearby stars with known distances, then we measure the Fluxes and determine the

Luminosities, to calibrate Luminosity classes in HlR diagram. In other words, one uses nearby stars

for which one can determine the stellar parallax and also the spectroscopic parallax .

Then for farther stars, knowing spectral class (or T) one can determine Luminosity. Next one

measure the Flux and get Distances (Spectroscopic Parallax).

The spectroscopic parallax is useful to determine distances within our galaxy

The Distance Ladder

With Newton’s modifications to Kepler’s laws, the period and size of the orbits yield

the sum of the masses.

P² = a³ /(m1 + m2 )

The relative distance of each star from the center of mass yields the ratio of the

masses.

m1d1 = m2d2

The ratio and the sum of the masses provide the information to calculate each mass

individually (Two equations and two unknowns). P, a, d1 and d2 are known (these four

parameters can be measured)

Note: For Sirius, the plane of the orbit is not face on, it is inclined 46 degrees from the line of sight. A

correction needs to be done first before using the values of size of orbit and distance to the center of mass

Stellar Masses:

Visual Binary Stars Binary star are classified as

visual, spectroscopic and

eclipsing The example shows Sirius (visual

binary), the brightest star in the sky.

Sirius A has a companion Sirius B, a

very dense object called white dwarf

Stellar Masses: Spectroscopic Binary Stars

In this example, using a telescope the observer cannot resolve the two stars and see the two stars as a single

star…

An example: The multiple star Castor. In a telescope one can see two stars. Each one of the two is a

spectroscopic binary. There is a third, fainter star in the Castor system which is also a spectroscopic binary

Many binaries are too far away or they orbit around the other star at a short

distance, but they can be discovered from periodic spectral line shifts. The shift in

wavelenght of the spectral lines as they orbit each other show a Doppler effect

Stellar Masses: Eclipsing Binary Stars

How do we identify eclipsing binaries?

We can identify an eclipsing binary by observing the light curve of the star, a plot of the apparent

brightness of the star as function of time

The occultation of the star in the system must be observed only if we can see the orbital plane

“edge on”.

This method also tells us something about the stellar radii (Through the deep of the eclipse).

The HR Diagram: Stellar Masses

Why is the mass of a star so important?

Together with the initial composition, mass defines the entire life

cycle and all other properties of the star!

The mass of a star will determine:

• Luminosity

• Radius

• Surface Temperature

• Lifetime

• Evolutionary phases

• And how the star will end its life….

All of this is determined by the mass of the

star. A note: The composition of the first stars was H and He. Heavier elements are

produced in the interior of the stars. After the interstellar gas was contaminated with

heavier elements produced in the interior of the first stars (Example: Supernova), the

composition of the mass of the later generation of stars incorporated those heavier

elements

Example: For stars on the Main Sequence, if

we plot the luminosity as a

function of mass, we find that the

luminosity depends of the mass (Notice that this plot is in log scale )

Luminosity Mass4

Why the luminosity increases

at such high rate?

A star with more mass means:

• more gravity

• more pressure in the core

• higher core temperatures

• faster nuclear reaction rates

• fast production of energy ( Mass4)

• higher luminosities!

• shorter lifetime

Lifetime Fuel available / How fast fuel is burned

Stellar lifetime Mass / Mass4 = 1 / Mass3

Or, since Luminosity Mass4

Stellar lifetime Mass / Luminosity

So for a star

(For main sequence stars)

How long a star lives is directly related to the mass!

Example: The Sun lifetime is estimated to be about 10 billion years.

A star with 10 times the mass of the Sun has an estimated lifetime of

10 million years!

Do the calculation!

Big (Massive) stars live shorter lives, burn their fuel

faster….

H-R diagram Location of stars of different masses

Stars of large mass will evolve fast and move off the main sequence faster that low mass stars

The turn off point of two open star clusters in

the H-R diagram showing their different ages

The turn off points: Points where the stars are moving

off the main sequence

Stars of higher mass leave the

Main Sequence earlier

What can we deduce from the

HR diagram and the turn off

points about the relative age of

these two clusters?

Cluster M 67 is younger than

NGC 188. Massive stars in M 67

are still in the main sequence.

Stars of similar mass in NGC

188 are off the main sequence