# Solutions to Quiz 2

```Solutions to Quiz 2
Point distribution within a question is just a guideline.
1. [3 points] Sketch a Hertzsprung-Russell diagram and label the following parts:
main sequence, red giants, white dwarfs, supergiants. Be sure to label the axes!
See slide 3 in the notes for class 17. Half a point each for labeling the main
sequence, red giants, white dwarfs, and supergiants correctly, half a point for
sensible axes (values not required), and half a point for overall correctness.
2. [4 points] Compare and contrast the life cycle of a star like our own to a star 10
times more massive. Be sure to explain why the life cycle diverges and what the
likely remnants will be. (Use both sides of the page if necessary.)
Both stars begin by contracting from a gas cloud [0.5pt] until they are hot and
dense enough for hydrogen burning to begin [0.5pt]. In both cases, when the core
hydrogen is expended (this takes far less time for the more massive star [0.5pt]),
helium burning begins and both stars enter the red giant phase [0.5pt]. The solarlike star is unable to burn the helium fusion products, so it eventually casts off its
outer envelope (forming a planetary nebula [0.5pt]) and ends up as an evercooling white dwarf [0.5pt]. The more massive star is able to burn heavier
elements in sequence, up until iron. Since iron is very stable, the star is no longer
able to support itself and explodes in a supernova [0.5pt], leaving a neutron
star/pulsar remnant [0.5pt].
3. [3 points] Describe how either pulsars or black holes were discovered. State what
part of the E/M spectrum was featured in the observations, and what physical
arguments were made to demonstrate that the objects were indeed very exotic.
[NO credit if both are described!]
Pulsars were discovered by the amazingly regular radio [1pt] pulses they emit
[0.5pt]. In some cases the frequency was so high [0.5pt] that the objects could
only be made of incredibly dense material, indicating that pulsars must be
spinning neutron stars [1pt].
Black holes were discovered as bright, variable X-ray [1pt] point sources [0.5pt].
Follow-up observations of one case revealed a massive star evidently orbiting an
unseen companion [0.5pt]. The inferred mass of the companion implied that it
must be a black hole [1pt].
BONUS [1 point]: A small object is said to escape from a larger one when its kinetic
energy &frac12; mv2 exceeds the magnitude GMm/r of its gravitational potential energy,
where M is the mass of the larger object and r is the initial separation. Use this fact to
carry out a “classical” derivation of the Schwarzchild radius of a black hole.
The trick here is to set v = c (the speed of light), equate the kinetic energy to the
gravitational potential energy magnitude, and solve for r, the Schwarzchild radius:
&frac12; mc2 = GMm/r [0.5pt].
Note that the m’s cancel, leaving:
r = 2GM/c2 [0.5pt].
```