PY 228: STELLAR ASTROPHYSICS

HOMEWORK #1: SOLUTIONS

1. A binary star system is observed containing a blue and a red star with roughly equal magnitudes. The optical spectra of the two stars are well approximated by black body curves with maximum intensity at a wavelength of 3250 Angstroms and 7250 Angstroms respectively. What is the ratio of the radii of these two stars?

Use the Wien law to calculate the surface temperatures of the two stars:

Then use the equation for the luminosity of a blackbody to calculate the ratio of radii:

2. If the two stars in the previous question produce a combined total flux at the Earth of 2.6x10**-6 erg/s/cm**2, and they have a measured parallax of 0.02 arcsec, what are the radii of the two stars?

First find the distance to the binary star system given the parallax:

Then compute the luminosity using the inverse square law (each star giving off half the total observed flux):

Then use the blackbody luminosity equation to compute each stellar radius:

3. Why is the Rayleigh-Jeans law so useful in radio astronomy?

The Rayleigh-Jeans law is an approximation to the Planck spectrum in the limit of low photon energy (hf << kT). In radio astronomy the photon energies are very low, ensuring this inequality and making the Rayleigh-Jeans law an accurate approximation.

4. Roughly how hot must a plasma be for most of the H to be ionized? For most of the He to be singly ionized? Doubly ionized?

In class we learned that interstellar hydrogen becomes collisionally ionized at a temperature around 10,000 K. If, instead, you consider the Saha equation, and note that the ionization will change when kT is of order the ionization potential, then you might guess that hydrogen becomes ionized at a temperature around 100,000 K. In either case, helium would become singly ionized when the temperature is about (24.6/13.6)=1.8 times higher, and doubly ionized when the temperature is about (54.4/13.6)=4 times higher.

5. Explain how the H-alpha absorption strength changes as we raise the temperature of a star.

From section 3.4.3 of your text, at low temperatures essentially all the hydrogen is neutral, and most of it is in the ground state. Since little H will be in the second state, there will be few chances for H-alpha absorption. The H-alpha line will therefore be weak in low temperature, K and M stars. As we go to moderate temperatures, most of the H is still neutral. However, more of the H is in excited states, meaning that a reasonable amount will be in level 2. H-alpha absorption is posible. As the temperature increases, the H-alpha absorption is seen to get stronger. At very high temperatures, the H becomes ionized. Since there is less neutral hydrogen, the H-alpha becomes weaker. This explains why the H-alpha line is strongest in middle-temperature stars, and dissappears in the hottest, O stars.