# Discussion 3: Properties of Stars

## Distance

Use trigonometric parallax (triangulation) for nearby stars.

Parallax = angle subtended (i.e., covered) by 1 A.U., the distance from Earth to the Sun. (1/2 of the diameter of Earth's orbit).

As distance increases the parallax angle decreases. Ultimately limited by the smallest angles measurable (resolved). The atmosphere limits resolution ability, must go to space --> Hipparcos satellite.

The distance at which 1 A.U. subtends 1" (one arc second) is called 1 parsec (pc). (parallax arc second).

dpc = 1/p(")

1 pc = 3.26 Light-years. Distance to nearest star, Alpha-Centauri:
p = 0.762" ---> d = 1.31 pc = 4.27 light-years

Best angle measurements of Hipparcos were down to p = 0.002" ---> d = 500 pc (1,630 light-years).

## Space Motion

1. Radial Velocity: the component of velocity along our line of sight (perpendicular to the plane of the sky). We measure this by the doppler shift of spectral lines in the spectra of the stars.  / 0 = v/c  < 0 --> Blueshift: moving toward us  > 0 --> Redshift: moving away from us

2. Proper Motion: the apparent motion of a star through the sky due to its Transverse velocity, vT, which is the component of a star's total space velocity across our line of sight.

measured as an angular speed (typically 0.1"/yr for nearby stars), but to convert to physical speed (km/s) need to know the distance

vT = 4.74 ("/yr)dpc

3. Spin: If a star is spinning (and they all seem to be) then, as it spins, one side of it is moving toward us while the other side is moving away from us. This will cause the width of spectral lines from its surface to be Broadened due to the doppler shift. The width of their broadening is  and so gives us the rate of spin of the star.

## Mass and Size

Direct measurements of mass and size of stars are not possible with our current Earth-bound technology. But we can infer these quantities with knowledge of physics and geometry if the stars are members of a binary system.
1. Binary Systems: Over 50% of the stars in the sky are actually multiple systems bound together by mutual gravity. Most are double. Some are visual binary systems: they can be seen in a telescope as two stars. Others are spectroscopic binary systems: only one star can be seen but the spectra is that of two stars that shifts back and forth from doppler effect. The two stars are in orbit about one another.

1. Kepler's 3rd Law: applies not only to planets in orbit around stars, but stars in orbit around a common center of mass. In binary systems we can measure the period of their orbits, and if we observe them carefully we can also get the size of their orbits (semimajor axis) by measuring their proper motion and parallax. Thus using Netwon's form of Kepler's third law we can measure the sum of their masses.

P2 = 4 2/G(m1+m2) * a3

2. Center of Mass: In order to get the individual masses of the stars we can make even more careful measurements of the binary orbit and determine each star's distance to their common center of mass. There is a simple relation between their distances from the center of mass and their masses:
m1a1 = m2a2
2. Eclipsing binaries: Sometimes there is an extraordinary alignment of a binary system to our line of sight and the two stars eclipse one another. When this happens the combined light from the two stars changes as one star passes in front of the other. There appear dips in the light curve. We can measure the stars' orbital velocities using the doppler shift of light and then use the timing of the eclipses to infer the size of each star.
(d = vt)

## Temperature

We can measure the temperature of the surface (photosphere) of stars by using their color. When we examine their energy spectrum we can see that they approximate blackbodies. Thus, we can can fit the mathematical spectrum of a theoretical blackbody over the observed spectrum and infer the effective surface temperature of the star using Wien's Law: peakT = 2.898x10-3 m*K

We can also infer the temperature of the surface from the specific spectral lines that are seen. Recall that spectral lines arise from transitions of an electron from one level to another in an atom. The electron may only emit or absorb very specific wavelengths of light for specific transitions. So, for example, say we see an absorption line of hydrogen which indicates that an electron has absorbed a photon and jumped from level 4 to level 7 in the atom. For the electron to already be excited enough to be in level 4 the atoms in the star's surface must be moving fast and hitting each other very often. This kind of motion is in fact how we define temperature. So this would be a pretty hot star.

## Luminosity

Luminosity is the total amount of energy that leaves a star per unit time. It therefore has units of energy per time. Watts is one such unit (joules/s). ergs/s is another.

Stefan-Boltzmann Law: Recall that we had a relationship between the temperature of a blackbody and how much energy per unit surface area per unit time it emitted. = T4

So if we want to know how much total energy per second a blackbody emits we need to multiply by the total surface area of the blackbody. Stars are spherical, so SA = 4 R2. Thus the luminosity, L, of a star is related to its radius, R, and surface temperature, T.

L = 4 R2 T4

Inverse Square Law of Light:
When you observe a star from a distance, d, you do not directly observe its luminosity because you only intercept a small fraction of the energy that left its surface. You observe the star's brightness, b. Brightness has unit of energy per unit surface area detected per unit time. Since light spreads away from a star in a spherical shell you can see that the amount of energy per unit surface area per unit time observed at a distance is just the Luminosity of the star divided by the surface area of a shell at the distance of the observer. This gives us the inverse square law of light.

b = L/(4 d2)

This says that if you move twice as far from a star (from some starting position) the star will become 4 times as dim.

It also says that if you know the distance to a star, and you measure its brightness, you can infer its luminosity. Then you can measure its temperature from spectrum and hence get its size.