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0. The Family of Stars. Chapter 9. What we can measure directly: Surface temperature and color Spectrum Apparent magnitude or intensity Diameter of a few nearby stars Distance to nearby stars. Part 1: measuring and classifying the stars. What we usually cannot :

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part 1 measuring and classifying the stars
What we can measure directly:

Surface temperature and color


Apparent magnitude or intensity

Diameter of a few nearby stars

Distance to nearby stars

Part 1: measuring and classifying the stars
  • What we usually cannot:
      • Distance to most stars
      • Luminosity (energy radiated per


      • Diameter and mass
surface temperature and color indices
Surface temperature and color indices

Color filters

Color indices:

B-V, U-B

Differences in apparent magnitudes observed

through different filters

spectral classification of stars
Spectral Classification of Stars

Mnemonics to remember the spectral sequence:

how to find distances to the stars
Parallax (only for stars within ~1500 ly)

From stellar motions

For moving clusters

Using “standard candles” (model-dependent)

Using mass-luminosity relation (for main-sequence stars) or period-luminosity relations (for binaries and variable stars; model-dependent)

How to find distances to the stars?

When d is too large, angles A and B become too close to 900

The larger the baseline, the longer distances we can measure


The longest baseline on Earth is our orbit!

Angular shift; we can measure it directly

Apparent shift in the position of the star: parallax effect


Larger shift

Smaller shift

Effect is very small: shift is less than 1 arcsec even for

closest stars

Aristotle used the absence of observable parallax to discard heliocentric system


The parallax angle p

Small-angle formula:

Define 1 parsec as a distance to a star whose parallax is 1 arcsec

1 pc = 206265 AU = 3.26 ly

d (in parsecs) = 1/p

the trigonometric parallax
The Trigonometric Parallax


Nearest star, a Centauri, has a parallax of p = 0.76 arc seconds

d = 1/p = 1.3 pc = 4.3 LY

With ground-based telescopes, we can measure parallaxes p ≥ 0.02 arc sec

=> d ≤ 50 pc

With Hipparcos satellite: parallaxes up to 0.002 arcsec, i.e. d up to 500 pc.

118218 stars measured!

proper motion
Proper Motion

In addition to the periodic back-and-forth motion related to the trigonometric parallax, nearby stars also show continuous motions across the sky.

These are related to the actual motion of the stars throughout the Milky Way, and are called proper motion.


Barnard’s star: highest proper motion

10 arcsec per year, or one lunar diameter per 173 yr

Approaches us at 160 km/sec

Fourth closest star

brightness and distance
Apparent magnitude: tells us how bright a star looks to our eyesBrightness and distance

Intensity, or radiation flux received by the telescope: Energy of radiation coming through unit area of the mirror per second (J/m2/s)

brightness and distance15



Brightness and Distance

The flux received from the star is proportional to its intrinsic brightness or luminosity (L) and inversely proportional to the square of the distance (d):





intrinsic brightness or luminosity
Intrinsic Brightness, or luminosity

The flux received from the star is proportional to its intrinsic brightness or luminosity (L) and inversely proportional to the square of the distance (d):



I =


Star A

Star B


Both stars may appear equally bright, although star A is intrinsically much brighter than star B.

brightness and distance18
Brightness and Distance



Recall the definition of apparent magnitude:

Define the magnitude scale so that two objects that differ by

5 magnitudes have an intensity ratio of 100.

Order of terms matters!


However, the apparent magnitude mixes up the intrinsic brightness of the star (or luminosity) and the effect of distance (which has nothing to do with the luminosity of the star).

Inverse square law:

distance and intrinsic brightness


Distance and Intrinsic Brightness


Recall that:


App. Magn. mV = 0.41


For a magnitude difference of 0.41 – 0.14 = 0.27, we find an intensity ratio of (2.512)0.27 = 1.28

App. Magn. mV = 0.14

distance and intrinsic brightness 2


Distance and Intrinsic Brightness (2)

Rigel is appears 1.28 times brighter than Betelgeuse,


But Rigel is 1.6 times further away than Betelgeuse

Thus, Rigel is actually (intrinsically) 1.28*(1.6)2 = 3.3 times more luminous than Betelgeuse.



Absolute Magnitude

A star that is very bright in our sky could be bright primarily because it is very close to us (the Sun, for example), or because it is rather distant but is intrinsically very bright (Rigel, for example). It is the "true" (intrinsic) brightness, with the distance dependence factored out, that is of most interest to us as astronomers.

Therefore, it is useful to establish a convention whereby we can compare two stars on the same footing, without variations in brightness due to differing distances complicating the issue.

Astronomers define the absolute magnitude M to be the apparent magnitude that a star would have if it were (in our imagination) placed at a distance of 10 parsecs (which is 32.6 light years) from the Earth.

To determine the absolute magnitude M the distance to the star must also be known!

absolute magnitude
Absolute magnitude

Recall that for two stars 1 and 2

Let star 1 be at a distance d pc

and star 2 be the same star brought to the distance 10 pc.


m2 = M


the distance modulus


The Distance Modulus

If we know a star’s absolute magnitude, we can infer its distance by comparing absolute and apparent magnitudes:

Distance Modulus

= mV – MV

= -5 + 5 log10(d [pc])

Distance in units of parsec


d = 10(mV – MV + 5)/5 pc


Absolute magnitudes of two different stars 1 and 2:

If two stars are at the same distance of 10 pc from the earth:

absolute magnitude 2


Absolute Magnitude (2)

Back to our example of Betelgeuse and Rigel:



Difference in absolute magnitudes: 6.8 – 5.5 = 1.3

=> Luminosity ratio = (2.512)1.3 = 3.3


Organizing the Family of Stars

We learned how to characterize stars with many different parameters

Is there any correlationbetween stellar

luminosities, radii, temperature, and masses???

the size radius of a star
The Size (Radius) of a Star

We already know: flux increases with surface temperature (~ T4); hotter stars are brighter.

But luminosity also increases with size:

Star B will be brighter than star A.



Luminosity is proportional to radius squared, L ~ R2.

Quantitatively: L = 4 p R2s T4

Surface flux due to a blackbody spectrum

Surface area of the star

example star radii
Example: Star Radii

Polaris has just about the same spectral type (and thus surface temperature) as our sun, but it is 10,000 times brighter than our sun.

Thus, Polaris is 100 times larger than the sun.

This causes its luminosity to be 1002 = 10,000 times more than our sun’s.


However, star radius is not a convenient parameter to use for classification, because it is not directly measured.

Surface temperature, or spectral class is more convenient!

organizing the family of stars the hertzsprung russell diagram
Organizing the Family of Stars: The Hertzsprung-Russell Diagram

We know:

Stars have different temperatures, different luminosities, and different sizes.

To bring some order into that zoo of different types of stars: organize them in a diagram of


Temperature (or spectral type)


Absolute mag.

Hertzsprung-Russell Diagram




Spectral type: O B A F G K M

hertzsprung russell diagram
Hertzsprung-Russell Diagram





Absolute magnitude

Sirius B

Color index, or spectral class


Check whether all stars are of the same radius:

Total radiated power (luminosity) L = T4 4R2 J/s

the radii of stars in the hertzsprung russell diagram


The Radii of Stars in the Hertzsprung-Russell Diagram



10,000 times the sun’s radius


100 times the sun’s radius


As large as the sun


Majority of stars are here

Specific segments of the main sequence are occupied

by stars of a specific mass


The mass-luminosity relation for 192 stars in double-lined spectroscopic binary systems.

L ~ M3.5much stronger than inferred from L ~ R2 ~ M2/3


However, this M3.5 dependence does not go forever:

Cutoff at masses > 100 M and < 0.08 M


H-R diagram for nearby+bright stars:

All stars visible to the naked eye + all stars within 25 pc