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.