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Astronomy 150
K. Nordsieck
Spring 2000
Solution
Homework 2: Due Feb 24, 2000, in class. (or turn it in before then to my mailbox, 5th floor,
Sterling Hall)
True or False (Circle T or F)
1. ( T F ) On the surface of a planet with twice the mass of the Earth, and twice its radius, you
would weigh less than you do on the Earth
True. F = m1 m2 / R2. m1 (you) is the same, m2 (the planet) is twice, but R2 (distance) is four
times.
2. ( T F ) Suppose you heat up two opaque objects of the same size so that one of them glows
red and the other glows yellow. The red one will be brighter.
False. By Wien's Law, since red is a longer wavelength than yellow, the surface temperature of
the red one must be smaller. But by Stefan-Boltzmann's Law, its luminosity/area must also
be smaller, and they are the same size. So the red one is less luminous.
3. ( T F ) Using parallax, astronomers can now reliably measure the distance of most of the stars
in our galaxy.
False. Sad to say, most stars are so far away that the parallax is not yet measurable. In the next
10-20 years, several proposed satellites may be able to do this.
Multiple Choice (circle a, b, or c)
4. At 1 A.U. the speed for a circular orbit around the sun is 30 km/sec. An object at 1 A.U. from
the sun moving at 25 km/sec is in a(n) _______ orbit.
A. circular
B. elliptical C. parabolic D. hyperbolic
5. Two stars of unequal mass orbiting each other under mutual attraction in circular orbits will
each circle around
A. the center of the most massive star
B. the half-way point between star centers
C. a point between the two stars, closest to the most massive star (definition of center of
mass)
D. a point between the stars, closest to the least massive star
6. Two stars that have the same surface temperature have the same
A. mass B. luminosity
C. radius
D. color (Wien's Law)
7. From a measurement of the heliocentric parallax and the flux of a star we can work out its
A proper motion
B radial velocity
C luminosity (parallax gives you distance; Lum = 4Bdist2 × Flux)
Short Answers
8. Sirius (" Canis Majoris, the very bright star to the lower left of Orion) is actually a visual
binary star consisting of Sirius A, a type A1 Main Sequence star, and Sirius B, a White Dwarf
companion. Since they are seen to orbit each other, they are at the same distance from the sun.
a. Sirius has a parallax of 0.38 arcsec. What is its distance in AU, parsecs, and light years?
Dist(pc) = 1/parallax(arcsec) = 1/0.38 = 2.63 pc
Dist(AU) = 206265 x Dist(pc) = 543,000 AU
Dist(ly) = 3.26 x Dist(pc) = 8.6 ly
b. The orbital period P of Sirius A and B is 50 years and the mean radius A of their orbit is 20
AU. What is the sum of their masses in Solar Masses (Msun)?
(MA + MB) / Msun = a(AU)3 / P(yr)2 = 3.2
c. Sirius A is half as far from the center of mass of the system as Sirius B. What are the
individual masses of Sirius A and B?
MA / MB = rB / rA = 1 / 0.5 = 2
MA = 2/3 (MA + MB) = 2.1 Msun
MB = 1.1 Msun
d. Sirius A has an apparent magnitude of -1.5 and the Sun has an apparent magnitude of -26.5.
What is the luminosity of Sirius A in Solar Luminosities (Lsun)?
Most astronomers tend to use the method of ratios, which avoids many hassles with the values of
constants and units. This problem is a good example.
We have the magnitude difference between Sirius A and the Sun, which corresponds to the flux
ratio. We need the luminosity ratio, so we use the inverse square law relating luminosity to flux
and distance.
1) Flux ratio (see formula on page 25 of the notes)
F(Sirius A) / F(Sun) = 10 -m(Sirius A)/2.5 / 10 -m(Sun)/2.5
= 10 -(m(Sirius A) - m(Sun))/2.5
m(Sirius A) - m(Sun) = -1.5 - -26.5 = 25
Flux(Sirius A)/Flux(Sun) = 10 -(25/2.5) = 10 -10
2) Luminosity ratio
Lum(Sirius A) = 4B Dist(Sirius A)2 × F(Sirius A)
Lum(Sun) = 4B Dist(Sun)2 × F(Sun)
Lum(Sirius A)/Lum(Sun) = Flux(Sirius A)/Flux(Sun) x (Dist(Sirius A)/Dist(Sun))2
= 10-10 x (543,000 AU/ 1 AU)2 = 29.5
Sirius A is about 30× more luminous than the Sun
e. Sirius B has apparent magnitude of 8.5. What is the ratio of the fluxes of Sirius A and B? of
the luminosities?
Just repeat the above method!
mag(Sirius B) - mag(Sirius A) = 8.5 - -1.5 = 10
Flux(Sirius B)/Flux(Sirius A) = 10 -(10/2.5) = 10 -4
Lum(Sirius B)/Lum(Sirius A) = 10 -4 (Distances are the same)