Measuring the Stars
Giants, Dwarfs, and the Main Sequence

# The Distances to the Stars

Stellar Parallax:
1/2 angle through which a star's position shifts as earth orbits the sun.
(actually this only works in determining stellar distances for nearby stars.
Table of nearest stars )
Nearest Stars:
• Alpha Centauri complex (triple-star system)
• Proxima Centauri at 1.3 pc (4.2 ly)
• 0.77 arc-secs
• Barnard's Star
• 1.8 pc (6.0 ly)
• 0.55 arc-secs

# Stellar Motion

• Stars are moving and the motion across the sky, after correction for parallax, is called proper motion
• The largest known proper motion of any star is that of Barnard's star (227 arc-seconds in 22 years)

• The true space motion is the combination of the transverse (proper) motion and the radial motion, determined from the Doppler shift of the stellar lines.

# Stellar Sizes

• Most stars appear as points of light, so their sizes cannot be directly measured
• for most we must use indirect means (temperature and brightness)
• Consider a "hot star" with surface temperature 30,000 K,
and a "cool star" with surface temperature 3,000 K
• These stars have names:
• Type O : 30,000 K
• Type M : 3,000 K
• What are their luminosities?
• We see this from a plot of the "main sequence" of the Hertzsprung-Russell Diagram
• So the Type O star is 1010 times as bright as a Type M star
• How does this compare to our expectations from Blackbody Theory?
• Stefan-Boltzmann Law says the energy emitted per unit area is proportional to T4
or emission per unit area from type O (30,000K) should exceed type M (3,000K) by:
104 = 10,000
but not by 1010 = 10,000,000,000 that we observe
we have an extra factor of 1,000,000 = 106
• So why are type O stars so much brighter?
• Because type O are also much bigger
Total brightness is proportional to
Area x T4, not just T4

or Luminosity ~ (Radius) 2 x T4

• So the extra factor of 1,000,000 = 106
represents an increased area, or a radius increase of 1,000
that is, type O has a radius 1000 times the radius of type M

The above equation can be rearranged to derive an equation for the radius:
(Radius)2 ~ Luminosity / T 4

or

## Radius ~ SQRT(Luminosity) / T 2

• look again at the radius trends on the plot of the "main sequence" of the Hertzsprung-Russell Diagram
• Stellar Sizes
• Stars can be classified by their size
• Giants (10-100 times the Sun's radius)
• Supergiants (100-1000 times the Sun's radius)
• Dwarf (comparable to or smaller than the Sun)

# Luminosity and Brightness

Luminosity:
Total rate at which radiative energy is given off by a celestial body

Brightness
The brightness that the star appears to have to an observer on the Earth
• This depends on how far away the object is by the inverse-square law:
• Knowing Brightness and Distance, we can determine Luminosity
• Brightness ~ luminosity / distance 2
• Often expressed relative to the sun's luminosity (LSUN).

Magnitude:
Measure of Brightness

Apparent magnitude:
Greeks (Hipparchus) established scale called Apparent magnitude
Apparent magnitude is a measure of Brightness
• Brightest stars visible to unaided eye = Magnitude 1
• Dimmest stars visible to unaided eye = Magnitude 6
• This is a logrithmic scale
• Measurements show 1st magnitude stars are 100x as bright as 6th magnitude stars.
• So, a Magnitude difference of 1 corresponds to a factor of 2.51 in brightness.
• or (2.51)5 = 100

Absolute Magnitude:
Apparent magnitude a star would have if it were exactly 10 parsecs from the Earth.
Absolute Magnitude = Luminosity, although in different units.

Luminosity and Brightness of the Sun
• Sun's Brightness = 1370 Watts/m2
• Sun's Distance (d) = 1.5 x 1011m
• Therefore, Area of Sphere is 4 (pi) d2
= 4 (pi) (1.5 x 1011m)2
= 3 x 1023 m2
• Luminosity= (1370 Watts/m2)(3 x 1023 m2)
= 4 x 1026 Watts

At a distance of 10 parsecs the Sun would be a magnitude 4.83 star
So the Absolute Magnitude of the Sun is 4.83

Another example: Sirius:
Brightest Star in the Sky
• Apparent Magnitude = -1.44
• Distance
• parallax = 0.38 arcsec
• distance (d) = 1/parallax = 2.5 parsec
• 3.26 l.y. in a parsec
• So, d = 8.6 l.y. (=2.5 x 3.26)
• Luminosity= 22 x Lsun
and Absolute Magnitude is 1.45

# Temperature and Color

Star's Surface Temperature
Our understanding of Blackbody Radiation explains why star's color is related to surface temperature.
Colors of Betelgeuse and Rigel in Orion are clearly red and blue
B (blue) and V (visual) filters admit different amounts of light for different temperatures
The color index, relating the B and V intensities, is defined in two ways
Star's absorption spectrum is also indicative of the surface temperature.
• Ionization state of atoms depends on temperature
• Energy of light (and therefore absorption) depends on temperature

# The Classification of Stars

Star's Absorption Spectrum
Used to classify Stars (according to Spectral Class)

Mnemonic for O B A F G K M :
Oh, Be A Fine Guy (Girl), Kiss Me

Types of Spectra
Hydrogen Lines Strongest in A spectra
Molecular Lines Strongest in M spectra
Neutral Metals Strongest in G,K, and M
Neutral Helium Strongest in B
Ionized Helium Strongest in O

The spectral class matches with surface temperature

Each lettered spectral class is further sub-divided in 10 subdivisions, denoted by 0-9
• so, for example:
• The Sun is G2,
• Betelgeuse is M2,
• and Barnard's star is M5

# Extending the Cosmic Distance Scale

Parallax only works for the closest stars

Spectroscopic Parallax
(not really a parallax - just a method for finding distance)
Consider "Main-Sequence" Star:
1. Find Spectral Type
2. Measure Brightness
3. Look at H-R diagram for expected Luminosity
And use inverse-square law to deduce distance

# Luminosity Classes

Only distances to stars on the main-sequence can be correctly determined by the simple procedure outlined above under spectroscopic parallax
since 9 out of 10 stars are on the main-sequence, this is already pretty good

Can we distinguish stars that are off the main-sequence by their spectra?
Yes

Technical Difference in Spectra of Same Spectral Class based on the width of lines

width of lines determine luminosity classes Ia-V

So now we have a third label for a star:
• The Sun is G2V, V being the stellar luminosity class for the main-sequence.
• Betelgeuse is M2Ia

An example of the discrimination between K2-type stars is illustrated in the following table:

# Stellar Mass

Stellar Masses and Binary Stars
Most stars are part of a multiple star system
From these measurements we can learn about masses:
• For visual binary use Kepler's law to deduce masses:
• M1 + M2 ~ a3 /p2
a = semimajor axis = radius for circular orbit
p = period of orbit
• Note: I don't expect you to know this formula, just that there is a relationship between the mass, the period, and the size (semimajor axis) of the orbit
• For other binaries, some information on orbits and masses can be derived
• Most of our knowledge of the masses of stars is based on these binary measurements

Example binary star: Sirius (the brightest star in the sky)
• bright Sirius A and faint companion Sirius B
• orbital period = 50 years
• semi-major axis = 20 AU
• MA + MB = 3.2 Msun
• further study reveals:
• MA = 2.1 Msun
• MB = 1.1 Msun

Stellar properites depend on the mass
The mass of the star at the time of formation determines its location on the main sequence.

Main sequence stars range in mass from 0.1 to 20 times the mass of the Sun (with a few exceptions)

Most main-sequence stars are low-mass stars, and only a small fraction are much more massive than the Sun

The main-sequence star's radius and luminosity depend on its mass

• Luminosity increases as (Mass) 3 for massive main-sequence stars
and (Mass) 4 for more common main-sequence stars
• Total fuel to burn in star is the mass
• Therefore:
............. More massive stars burn up fastest and have shortest lives

• and since the luminosity increases as the cube of the mass for the most massive stars:

• (these are only approximate relationships)

# Star Clusters

Star clusters are a collection of stars at approximately the same distance from us, so they can be compared without correcting the brightness to absolute brightness (or we don't need to know how far away they are to compare them)

Open (or Galactic) Clusters
a loose association of young stars - metal-rich
• we assume these stars were formed around the same time
• O type stars are still on the main-sequence, therefore this cluster is younger than lifetime of O type stars: 25 million years
• heavy elements are abundant in these stars
• metal-rich
• created by earlier stars

Globular Clusters
large spherical clusters of stars- metal-poor
Example: Omega Centauri
• again, we assume these stars were formed around the same time
• no O or B type stars
• no main-sequence stars with masses greater than 0.8 solar masses
• A type stars are passing back through the main-sequence
• few heavy elements are found in these stars
• metal-poor
• so stars were created in distant past, when heavy elements were less abundant
• We can conclude that this (and other) globular cluster was formed over 10 billion years ago
• recall lifetime of the Sun will be 10 billion years, and if it had formed with this cluster is would have been extinguished by now, as seen on the H-R diagram

• Open clusters are of recent (last billion years) origin
• Globular clusters are very old (more than 10 billion years)