Solutions to Homework 3
1. Referencing the internet or any other suitable resource, write a paragraph
describing one of the following three topics:
a. The solar neutrino problem.
b. Supernova 1987A.
c. The GAIA mission.
Be sure to describe why the topic is interesting, and, if applicable, report on any
recent developments related to the topic.
For (a), early measurements of solar neutrino flux were smaller than expected by
a factor of 3. This was critical because the observations seemed at odds with
stellar theory. But recently it has been determined that neutrinos can change
between three forms with roughly equal probability. The early experiments were
only sensitive to one form, hence the factor of 3 difference.
For (b), SN1987A was significant because it was the first time neutrinos had been
definitively detected from a supernova, yielding important information about the
process. But there were also mysteries: the progenitor star was apparently a blue
giant, which theory suggested should not explode, and astronomers have still been
unable to locate the expected pulsar at the center of the nebula.
For (c), the GAIA mission is described at http://sci.esa.int/gaia. In short, the
mission will map the positions and motions of a large fraction of all stars in the
galaxy to high precision, yielding new information about the structure of the
Milky Way and refining the cosmological distance ladder. The mission should
last 5 years and is scheduled to launch in 2010.
2. The starship Nemesis enters orbit around a distant blue star. Use the following
equations to work out the problems below:
Newton’s form of Kepler’s 3rd law: (Distance)3  Mass  (Period)2 .
Stephan-Boltzmann law: Luminosity  (Radius)2  (Temperature)4 .
a. It takes Nemesis 2 yr to orbit the star at a distance of 5 AU. What is the
mass of the star, in units of our Sun’s mass (M)? Show your work.
Use the fact that it takes the Earth 1 yr to complete its 1 AU radius orbit
around the Sun. Expressing Newton’s form of Kepler’s 3rd law in terms of
ratios, we find (5AU/1AU)3 = (M/1M)(2yr/1yr)2, giving M= 31 M.
b. Nemesis measures the star’s radius to be 10 R (i.e., 10 times larger than
one solar radius) and its surface temperature to be 9.5 T. What is the
star’s luminosity? Show your work. Is your answer reasonable
considering what you found for the star’s mass in part a? Hint: most stars
follow the “mass-luminosity” relation, Luminosity  (Mass)4.
From the Stephan-Boltzmann law, (L/L) = (10 R/1 R)2(9.5 T/1 T)4,
or L = 8.1  105 L. From the mass-luminosity relation, we would
expect the luminosity to be ~314 L = 9.2  105 L, which is close.
c. What is the likely fate of this star? I.e., will it become a white dwarf,
neutron star, or black hole?
Since the star’s mass is above 20 M, the star will likely become a black
hole after it explodes in a supernova.
3. Estimate the expected lifespan of the Sun, assuming all its hydrogen is consumed
by fusion and that its luminosity (energy released per unit time) is constant at L
= 3.85  1026 W (in reality, the Sun’s luminosity increases over time as it burns
hydrogen in its core). Show your work and express your answer in Gyr. Use the
fact that the Sun started with roughly 1.5  1030 kg of hydrogen and that for every
kg of hydrogen consumed during fusion, 0.7% is liberated in the form of pure
energy via the equation E = mc2, where m is the mass and c = 3  108 m s-1 is the
speed of light. Your answer should be quite a bit larger than the Sun’s actual
lifetime of about 10 Gyr. What do you suppose is the reason for this difference?
The lifetime is the total available energy divided by the luminosity, or t = 0.007 
1.5  1030  (3  108)2 / 3.85  1026 = 2.45  1018 sec = 78 Gyr. This is the time it
takes to burn all of the Sun’s hydrogen. But the nuclear reactions only occur in
the Sun’s core, so not all of the Sun’s hydrogen participates in the nuclear fusion.
In fact, only about 1/8th of the Sun’s mass participates… hence the lifetime is
shorter by a factor of ~8.