Download Cosmology 20B Homework 2 solutions

Cosmology 20B Homework 2 solutions
1. What are absorption lines, what causes them, and how are they used by astronomers to figure out what
astronomical objects are made of?
Absorption lines are regions of very low light (dark spots) in a galaxy’s spectrum that are
caused by gas that is sitting between the observer and the light source. Atoms tend to absorb
light at specific wavelengths in association with their electron energy levels. By categorizing the
absorption lines astronomers can figure out what types of atoms (and in what proportion) make
up the gas.
2. Earth’s atmosphere shields us from most kinds of electromagnetic radiation. Name two kinds of electromagnetic radiation (e.g. gamma-rays, UV, etc.) that do reach the Earth’s surface.
Visible light and Radio waves reach the Earths surface through the atmosphere.
3. Are we more likely to want to put a gamma-ray telescope into space or a radio telescope into space? Why?
We are more likely to put a gamma-ray telescope into space because gamma rays from astronomical objects do not reach the ground because they are absorbed by the Earths atmosphere.
Radio telescopes work just fine on the ground because radio waves penetrate the atmosphere.
Also, radio telescopes must be very large (because radio waves themselves are so big). Launching
a radio telescope into space would be extremely expensive.
4a. What is black body radiation?
Black body radiation describes the type (or spectrum) of radiation that is emitted by an
object that is perfect, in the sense that it reflects no light, and only emits radiation because of
its temperature.
4b. You are looking at two hot objects that are glowing because of their temperature. One glows blue and
the other glows red. Which one is hotter? Explain how you know this using the Wien’s law equation for
black body radiation.
Wien’s law tells us that a black body will radiate more of its energy at shorter wavelengths
the hotter it is. Blue light has shorter wavelength than red light, so the blue-glowing object is
hotter.
5. What does E = mc2 mean in words? What does it have to do with stars?
E = mc2 means that there is some amount of energy associated with every piece of mass,
and it is equal to that mass times the speed of light squared. In this sense, energy and mass are
equivalent, because mass can (in principle) be converted to energy and vice versa. The energy
source of stars comes from this type of energy, as four H atoms fuse to form a (less massive) He
nucleus.
6. What is a black hole?
A black hole is an object that is so dense, with a gravitational field is so strong, that light
cannot escape. In a sense, a black hole is an object with an escape speed larger than the speed
of light.
7 .Under what circumstances does a supernovae end up as a black hole ?
A supernova will retain a black hole if its core remains more massive than 2 to 3 solar masses.
8. What two forces are competing when a star is in “hydrostatic equilibrium”?
Gravity pulling in and pressure pushing out compete when a star is in hydrostatic equilibrium.
9. (Will be graded) Hint: use the convenient rewrite of the E = mc2 equation that already gives you E in
Joules that I gave you in Lecture 7.
A. According to the E = mc2 equation, how many Joules of energy are contained in 10 grams of sugar?
E = 9 × 1013 Joules
m
g
= 9 × 1013 Joules
10g
g
= 9 × 1014 Joules
B. According to E = mc2 , how many Joules of energy are contained in 10 grams of fat?
This was a trick question. Same as A. above.
C. Our bodies can extract about 9 Food Calories of energy from each gram of fat. How many Joules of
caloric energy can our bodies extract from 5 grams of fat? Assume that a Food Calorie is equal to 4, 200
Joules.
= 9 KCal 5g
= 45 KCal (4200Joules/KCal) = 1.89 × 105 Joules
E = 9 KCal m
g
g
One of the following (10-12) will be graded:
10. If two stars have the same intrinsic luminosity, but one star looks sixteen times dimmer than the other,
how many times farther away is the dimmer star than the brighter star?
The equations you need are b1 =
L1
4πR12
and b2 =
L2
4πR22
Use L1 = L2 = L and b1 = b2 /16. What is R1 in terms of R2 ?
b1 =
L
4πR12
=
b2
16
=
1
L
16 4πR22
Matching the second and fourth terms implies that 16R22 = R12 or that R1 = 4R2 . Thus the
dimmer star is four times farther away.
11. What is the apparent brightness b of a nearby star with exactly the same intrinsic luminosity, L, as the
sun, but which is 4 light years away? (Hint: use b = 1370 watts/m2 for the brightness of the sun.)
Start with the apparent brightness of the Sun:
b =
L
2
4πRsun
← Equation 1
2
The distance to the sun is Rsun = 1AU = 1.5 × 101 1m. We are given b = 1370 Watts/m .
Now, consider a star at a distance, R∗ = 4 lt yrs = 4 × 9.5 × 1015 m = 3.8 × 1016 m. The
distant star has the same luminosity, L∗ = L and therefore has an apparent brightness:
b =
L
2
4πR∗
← Equation 2
Now, the only that we dont know in calculating b? is the value of L?. But we can solve for
2
. Plugging this into Equation 2 for b∗ we have:
that using Equation 1 above: L = b 4πRsun
b =
L
2
4πR∗
=
2
b 4πRsun
2
4πR∗
= b
2
Rsun
2
R∗
= 1370W/m2 (1.5 × 1011 )2 /(3.8 × 1016 )2 = 2.1 × 10−8 W/m2
12. You are 2m away from a candle and 5m away from a 75 watt light bulb. Both the candle and the light
bulb look to be the same brightness to your eye. How many watts of power is the candle producing?
The brightness falls off like the distance squared so we have bcandle = 75W (2m/5m)2 = 12W
The following 3 questions have no precisely correct answers but were graded on strength of argument
and overall understanding.
13. How did the Danish astronomer, Ole Romer, argue that light moved at a finite speed and provide an
estimate for the speed of light during in early 17th century? Explain the chain of argument. You are allowed
to draw a picture if you’d like.
14. Discuss the controversy between Darwin and Kelvin over the age of the Sun. What were the arguments
behind each of their views and who was right in the end? Taking into account Darwin’s authoritative view
on the timescales required for evolution, do you think that life in the Universe would be possible if energy
could not be produced from mass (E 6= mc2 ) and there was no source of nuclear energy in massive objects
like the sun? Why? (∼ 1 - 2 paragraphs)
15. Describe how the ancient Greeks (Aristarchus) determined the approximate distance to the Moon. Be
sure to describe the logical chain of reasoning used.