Addendum 3: Properties of Stars (Chapter 16)

First define some constants and dimensional units needed below

1. Random walk and escaping photons. An object that takes randomly oriented steps of length z will be, on average, adistance L from the starting point after n steps, where n is given by (derivation in lecture notes):

If the object moves at speed V, the time t to reach a distance L is:

Example: How long do photons created in the core of a star of radius 10x the Sun take to reach the surface of the star? Assume a mean free path between collisions of 1mm.

2. Stellar Parallax. The relationship between aa star's annual apparent angular motion w.r.t .distant star field [q, measured in arcseconds] and its distance d, [measured in parsecs (pc)] is

Example: A star has an observed parallax of 0.043" What is its distance in parsec?

3. Stellar Magnitude scale. The stellar magnitude scale is logarithmic with base 2.5. Hence, to compare the brightness of two stars with an apparent magnitude difference Dm, raise the base 2.5 to the Dm power.

or

Note:

Example 1: A double star system is observed at apparent magnitudes 5.7 and 8.3. Which one is brighter, and by much much (i.e., what is the brightness ratio?)

Note: the star with magnitude 5.7 is brighter!

This is the brightness ratio.

Example 2: Two stars differ by a factor of 100 in brightness. The fainter star is apparent magnitude 14.2. What is the apparent magnitude of the brighter star?

4. Kepler's 3rd law. The relationship between the component masses M₁, M₂ (kg) in a binary system, the orbital period P (seconds), and the semi-major axis a (meters), is:

This equation can be written is more convenient form by a using particular set of dimensions: mass in solar masses, period in Earth years, and a in AU.

Example 1: An eclipsing binary system is observed to have a period of 10 days, and maximum separation of 0.02 arcsec and a parallax 0.1". What is the sum of stellar maases in Solar mass units?

First we will do the problem using the first form of the equation:

distance calculated from the parallax

semi-major axis of system calculated using the small angle formula for distance and size

period in seconds

or

Now try using the second form using AU and years for dimensions:

4a. Kepler's 3rd law applied to eclipsing spectroscopic binaries. Spectroscopic binaries are stars in wwhich the speed of one or both components can be determined by the periodic Doppler shifts of their spectral lines as illustrated below. If the systmeis an eclipsing system, the orbital plane is parallel to the line of sight, so that the measured speeds are the true orbitial speeds. This can be used to determine the semi-major axis, since:

If both speeds can be measured, the mass ratio can be determined:

(follows from momentum balance)

Since from Kepler's 3rd law, the sum of masses can also be computed, the individual masses can be determined.

Example: Calculate the masses of each component of the eclipsing binary system shown below.

From the plot (note that the system is moving away from the observer at 40 km/s, which we subtract from both maximum velocities):

Hence

5. Luminosity as function radius, temperature. We have used this equation before, but it's worth repeating here:

Example: What is the radius of a star (in solar radii) with a surface temperature of 4500 K and luminosity 100x that of the Sun?

6. Distance, apparent magnitude, absolute magnitude. The relationship between apparent visual magnitude m_v , absolute visual magnitude M_v, and distance d (pc) is:

or

Example: The star has an absolute magntude M_v = -7.2 and an apparent magnitude m_v = +1.2. What is the distance to Deneb in parsec?

6. Measuring stellar distances using 'spectroscopic parallax. If a star is known to be on the main-sequence, there is a definite relationship between the absolute magnitude and spectral type. This allows us to determine the absolute magnitude of a star by measuring its spectral type and using the HR diagram, as illustrated below.

Example: A star is observed to have a spectral type (from its spectral lines) of F2V (recall that luminosity class V => main sequence). Its apparent visual magnitude is 9.5. What is the distance to the star?

From the HR diagram (see red arrows) a main-sequence F2 star has an approximate absolute magnitude:

Hence