Download Phys133 Mid-Term#2 Short problem samples 1. A star has a radius

Phys133 Mid-Term#2
Short problem samples
1. A star has a radius R=5 Rsun and temperature T=2 Tsun.
a. What is the star's luminosity L in units of Lsun?
L/Lsun = (T/Tsun)4 (R/Rsun)2 = 52 24 = 25 x 16 = 400
b. What the ratio of the star’s surface area to that of the sun,
A/Asun.
A/Asun = (R/Rsun)2 = 52 = 25
2. A main-sequence star with twice the sun's mass has a luminosity that
is 8 times the solar luminosity. Compared to the sun's main sequence
lifetime of 10 Byr, about how long can this star live on the main
sequence?
t = 10 Byr (M/Msun)/(L/Lsun) = 10 Byr 2/8 = 2.5 Byr
3. Star A and B are both on the main sequence with the same spectral
type, but Star B is observed to have 9 times the apparent brightness
of star A.
a. What is the ratio of the luminosity of star B to A, i.e. what is
LB/LA?
Same Spectral type on MS => same luminosity => LB/LA =1
b. What is the ratio of the observer's distance from star B to star
A, i.e what is DB/DA?
9= FB/FA = (LB/LA)/( DB/DA )2 => DB/DA = 1/Sqrt[9] = 1/3
4. A binary star system has a parallax of p=0.05 arcsec.
a. What is the system's distance d in parsec (pc)?
d/pc = 1/(p/arcsec) = 1/0.05 = 20 => d =20 pc
b. If the star's orbit in a plane perpendicular to our line of sight with
an angular separation a=0.5 arcsec, what is their physical
separation s in AU?
s/AU = (a/arcsec) (d/pc) = 0.5 20 = 10 => s = 10 AU
5. A star with radius R=5 Rsun has a surface temperature T=12,000 K.
a. What is the ratio of the star's temperature to that of the sun,
T/Tsun?
Tsun = 6000 K => T/Tsun = 12,000/6,000 = 2
b. What is the ratio of the star's surface area to that of the sun,
A/Asun?
A/Asun = (R/Rsun)2 = 52 = 25
c. What is the ratio of the star's luminosity to that of the sun,
L/Lsun?
L/Lsun = (T/Tsun)4 (A/Asun) = 24 52 = 16 * 24 = 600
6. In kilometers (km), what is the radius Rbh of a black hole with a mass
10 Msun?
Rbh = 3 km (M/Msun) = 10 * 3 km = 30 km
7. A star has its spectral energy peak at wavelength λmax = 250 nm.
a. What is the star's surface temperature T in Kelvin?
λmax = 500 nm (Tsun/T) = 250 nm =>
T = 2 Tsun = 2*6000 K= 12,000 K
b. If the star has a radius R=2 Rsun, what is its luminosity L, in
Lsun?
L/Lsun = (R/Rsun)2 (T/Tsun)4 = 22 24 = 64 => L= 64 Lsun
8. Two stars of equal mass orbit each other over a period P= 4 years
with a fixed orbital separation a=2 AU.
a. What is the mass of each star?
Kepler's 3rd law for binary:
(M1 + M2)/Msun = (a/AU)3 / (P/yr)2 = 23/42 = 8/16 = 1/2
But M1=M2, so M1= (1/4) Msun = M2
b. b. What is the orbital speed Vorb of each of these stars, in
AU/yr?
Separation a=2 AU is the diameter of each star's orbit, and orbital
circumference is π*diameter = π a. Orbital speed (distance/time) is
Circumference/Period, or
Vorb = π a/P = 2 π AU/4 yr = (π /2) AU/yr
c. Given that the earth's orbital speed is 30 km/s, what is the
orbital speed Vorb in km/s?
Vearth = 2 Pi π /yr = 30 km/s => Vorb = Vearth/4 = 30/4 = 7.5 km/s
9. Star A is a solar "twin" (with same Mass, Radius, etc. as the Sun), but
has a parallax of 0.1 arcsec. Star B has the same apparent
brightness and color as A, but a parallax of 0.02 arcsec.
a. What are the distances of A and B, in parsec (pc)?
d/pc = 1/(a/arcsec) => dA=1/0.1 = 10 pc ; dB=1/0.02 = 50 pc
b. What is the luminosity of star B, in units of Lsun?
same app. Brightness => FA=FB so use F=L/(4π d2) and solve for L:
LB/LA = (dB/dA)2=(50/10)2=25, but since A is solar twin, LA=Lsun =>
LB = 25 Lsun
c. About what is the surface temperature of star B, in degrees K?
Same color as A, which is solar twin, so TB=Tsun=6000 K.
d. What is the radius of star B, in units of Rsun?
LB/LA = (TB/TA)4 (RB/RA)4, but since TB=Tsun and RA=Rsun,
RB/RA=Sqrt[LB/LA] = Sqrt[25] => RB =5 Rsun