Download Midterm 1 Solutions

Midterm 1 Solutions
Astronomy 160 Fall 2006
1.
Problem 1
A main-sequence star of spectral type B0 has an absolute magnitude Mbol of −4.1, and an effective
temperature Tef f of 28,000 K. It has an apparent bolometric magnitude, mbol of +5.9.
The absolute bolometric magnitude of the Sun is Mbol = +4.75 .
(a) What is the distance D of the star in pc?
mbol = Mbol + 5 log(D/10 pc)
5 log(D/10) = 10 → D = 1000 pc
(b) What is the star’s luminosity L∗ in solar units?
Mbol = −2.5 log(L∗ /L ) + 4.75
2.5 log(L∗ /L ) = 4.75 − Mbol = 8.85 → log(L∗ /L ) = 3.54
L∗ = 3470 L
(c) Suppose it turns out that the “star” is actually a binary, with components whose luminosity
ratio is 2:1. What are their individual apparent bolometric magnitudes?
The bright star has L1 = 2 L2
L1 = 2/3 L∗ = 2310 L → log(L1 /L ) = 3.36
L2 = 1/3 L∗ = 1156 L → log(L2 /L ) = 3.06
(Mbol )1 − (Mbol ) = −2.5 × 3.36 → (Mbol )1 = −3.65
(Mbol )2 − (Mbol ) = −2.5 × 3.06 → (Mbol )2 = −2.90
mbol = Mbol + 10
Hence:
(mbol )1 = +6.35
(mbol )2 = +7.10
–2–
2.
Problem 2
Within interstellar space, there exists regions of ionized gas (“HII regions”), with typical temperatures and number densities of T = 8000 K and n = 100 cm−3 . Consider a spherical HII region,
composed of pure hydrogen, with a diameter D = 5 pc. Surrounding the region is relatively cold
gas.
(a) Estimate numerically (in ergs) Eint , the region’s total internal energy. Assume that the region
is isothermal and has uniform density.
We may assume from the statement of the problem that the gas is ideal and monatomic (and
thus has three degrees of freedom).
Z
3 RT
3
P dV =
M
Eint = Etherm =
2
2 µ
M=
4π D 3
( ) ρ0 = 1.47 × 1035 g
3 2
Eint = 2.9 × 1047 erg
(b) Calculate the Jeans length, λJ , in pc. Given the fact that the HII region is neither expanding
nor contracting, what are the two opposing forces keeping it in this equilibrium?
λJ =
πa2T
Gρ0
a2T =
1/2
RT
µ
µ = 0.5 (f ully ionized hydrogen gas)
aT = 1.2 × 106 cm/s
ρ0 = µmH n = 8.30 × 10−23 g/cm3
λJ = 8.7 × 1020 cm = 290 pc 5 pc
Since D λJ , the region would expand in the absence of external forces. The two main opposing
forces are the internal pressure of the gas and the external pressure from the surrounding cold gas.
The self gravity of the HII region is negligible.
(c) Estimate numerically (in ergs) Egrav , the gravitational potential energy. Is this energy positive
or negative.
Using R = D/2 = 7.5 × 1018 cm,
Egrav ≈ −
GM 2
= −1.9 × 1044 ergs
R
–3–
3.
Problem 3
A star of mass M∗ and radius R∗ is embedded within a large, diffuse gas cloud. This gas is pulled
inward by the star’s gravity, and settles into a state of hydrostatic equilibrium. Assume that the
only force of gravity is from the star, i.e., ignore the self-gravity of the gas cloud.
(a) Write a differential equation for P (r), the pressure in the cloud as a function of distance
from the star. In your equation, let ρ be the cloud’s mass density.
Simply state the equation of hydrostatic equilibrium:
dP
ρGM∗
= −ρg(r) = − 2
dr
r
(b) Assume the cloud is isothermal, with an isothermal sound speed aT . Let ρ∞ be the
unperturbed cloud density, i.e., the value at great distance from the star. Find ρ(r), the cloud
density as a function of distance.
Isothermality implies that P/ρ = a2T (a constant).
dρ
ρGM∗
=− 2
dr
r
Z ∞
Z ∞
dρ
GM∗
=−
dr
ρ
a2T r2
r
r
ρ∞
GM∗ 1 ∞
GM∗
ln
= 2
=− 2
ρ
r r
aT
aT r
a2T
ρ = ρ∞ exp
GM∗
a2T r