ASTR1030 Homework 3 by maclaren1


                                            ASTR1030 Homework 3
                                 DUE WEDNESDAY, OCTOBER 7 IN CLASS.
General Instructions For Homework:
• You may study with your friends but the work you hand in should be your own. The main purpose of the homework is for
   you to learn material on which you will be tested. You cannot learn if others do the work for you.
• Write your answers on this homework page. Add a regular 8 1/2” x 11" paper if necessary. Please staple and don't forget
   your name!
• A single number or word is not an answer. You must show how you came to your answer.
• Please write neatly, use appropriate units, and include the minimum number of significant figures to answer the question.

1. Angular Momentum
Even though all objects attract each other through gravity, astronomical bodies rarely collide since they have angu-
lar momentum with respect to each other. Angular momentum is a conserved quantity defined as L = r × mv .
Let’s see what happens if we try to move an object orbiting the Sun at 1 AU toward the Sun.
(2 pts) Just for reference, suppose we had a 1 kg object on a merry-go-round, 5 m from the center, and traveling at
2 m/s. What is its angular momentum? Express your answer in kg m2/s.
(3 pts) What is the angular momentum of a 1 kg object orbiting the Sun in a circular orbit at 1 AU. What is it’s
orbital speed? You will notice quite a difference between the two answers. What do you conclude?
(3 pts) Suppose we look at the same 1 kg object orbiting at 1/4 AU (just inside of Mercury). What is it’s orbital
speed? What is its angular momentum? Is it less or more than the object orbiting at 1 AU?
(2 pts) Folsom stadium has a mass of roughly 107 kg, most of which is ~100 m from its center. How fast would we
have to spin the entire stadium to duplicate the angular momentum of a 1 kg object orbiting the Sun at one AU?
What do you conclude? How easy is it to move an object closer to or farther from the Sun?

                                        ASTR1030 Fall 2009. Homework 3. Page 1
2. Thermal Emission
Calculate the wavelengths of peak emissions from the following objects (nm). Next calculate the wave's frequency
(Hz) and the photon energy (answer in eV; 1 eV = 1.6x10-19 J). What type of light is it (visible, infrared, radio, etc.)?
(2 pts) A hot stove (~300 oC). Hint: convert to K first!
(2 pts) A hot white dwarf star (29,000 oK).
(2 pts) The surface of the Sun (5800 oK).
(2 pts) A comet in the Oort cloud (10 oK).
(2 pts) The region near a black hole, called the accretion disk (~5,000,000 oK).

3. Glowing
The human body’s temperature is 37 oC.
(2 pts) At what wavelength will maximum emissions be? Answer in nm. What type of light is it?
(2 pts) How much power per unit area does the human body emit? Answer in W/m2.
(1 pt) The surface area if a human is roughly 2 m2. How much total power does the human body emit?
(2 pts) Convert the above answer into cal/hr. (calories/hour). Hint: 1 J = 0.25 cal. Don’t forget to convert seconds
into hours.
(1 pt) Keeping in mind that 1 Cal (food) = 1000 cal (physics), how many calories per day must a human eat just to
make up for radiated heat? Does this number seem too high or too low. Why don’t we all freeze?

                                        ASTR1030 Fall 2009. Homework 3. Page 2
4. Arcturas
(3 pts) Arcturas (Boötes) is one of the fastest moving stars in our sky at 118 km/s. The most dominant hydrogen
emission line (called Lyman α) is measured in the Laboratory to be at 121.6 nm. How much will light at this wave-
length be Doppler shifted if it is coming from Arcturas? Assume that it is moving towards us. Does this star’s
motion significantly change its color as seen by the human eye? (For example, could a red star appear to be blue do
to motion at this speed).
(2 pts) Most galaxies, however, are moving away from us. Suppose a galaxy is moving away at 104 km/s. What is the
Doppler shift for Lyman α? Is it a red shift or blue shift?

5. Extrasolar Planets
Most extra-solar planets have been detected using Doppler shifts. With this method, one must make an extremely
accurate measurement of the star’s velocity. The motion of a star induced by a large planet is ~100 m/s. One must
resolve 10 m/s to make this measurement (the best we can do).
(3 pts) What is the Doppler shift in a 500 nm emission from a star moving at 10 m/s?
(2 pts) How accurately must we measure Δλ, that is, what is Δλ/λο?

                                      ASTR1030 Fall 2009. Homework 3. Page 3
6. The Solar Corona
The Sun has a sphere of hot gas surrounding it called the corona. We measure peak emissions form the corona at
2.9 nm.
(1 pt) What type of light has a wavelength of 2.9 nm?
(2 pts) How much energy is in a photon at that wavelength? Convert your answer to eV.
(2 pts) What is the temperature of the corona?

7. Binary Systems
Over half of stars are in binary systems, only some of which can be resolved with telescopes. αcrux (Crux), one of
the brightest stars in the southern sky and a multiple star system, is observed to have 0.01 arcsec parallax motion.
Two stars are visible and separated by 4 arcsec.
                                                                                               Star A


                             4 arcsec                b
                                                                                                  Star B
(2 pts) How far away are the αcrux stars (What is b)? Express your answer in parsecs. Convert into ly and into m.
(2 pts) Using the diagram above, how far apart are the stars? (What is a)? Answer in m and AU.
(3 pts) How large a telescope is needed to resolve this binary system in visible light (550 nm)?

                                        ASTR1030 Fall 2009. Homework 3. Page 4
8. Lost at Sea
During a spring break, you decide to take a solo boat trip. While contemplating the universe, you drop your GPS
overboard and lose track of your location. Fortunately, you have some instruments, as well as a UT clock. You put
together the following description of your situation:
• It is the spring equinox.
• The Sun is on your meridian at altitude 75o in the south.
• The UT clock read 22:00.
(2 pts) What is your latitude? How do you know?
(3 pts) What is your longitude? How do you know?
(3 pts) Consult a map. Based on your position, where is the nearest land? Which way should you sail to reach it?

9. Telescope Resolution
A star one parsec away will have one arc second of parallax motion. The nearest stars are all more than one parsec.
away and even have less parallax motion, so we need about 0.5 arcsec resolution to even detect parallax motion
from the closest stars.
(3 pts) How large a telescope is needed to resolve 0.5 arcsec in visible light (500 nm)? Does this explain why dis-
tance to stars has only recently been determined? What else was needed to detect parallax motion?
(2 pts) In reality, we would like to have 10-3 arcsec (milli-arc second) to determine the distance to ~1000 nearest
stars. Even with this resolution, we cannot detect any parallax in vast majority of the stars in the Milky Way. How
large a telescope is needed to resolve 10-3 arcsec? Do we have such large telescopes?

                                      ASTR1030 Fall 2009. Homework 3. Page 5
10. Luminosity
An object’s luminosity can be thought of as the total power radiated from it. The Sun’s surface temperature is 5776
(3 pts) How much power per unit area does the Sun radiate? This value is known as LSun, the solar luminosity.
(2 pts) The Sun’s radius is 7x105 km. How much total power does the Sun radiate?

11. Sirius
Sirius, the brightest star in the sky, appears blue. The wavelength of maximum intensity, λmax, is about 290 nm.
(3 pts) What is the surface temperature of Sirius?
(3 pts) How much power per unit area does Sirius radiate?
(2 pts) Sirius’ radius is 1.71 that of the Sun. How much more total power does Sirius radiate?
(2 pts) The above value is known as luminosity. Typically, luminosity is expressed in solar luminosities. How many
times the Sun’s luminosity is Sirius’s? Use LSun = 3.85x1026 W.

                                      ASTR1030 Fall 2009. Homework 3. Page 6
12. Aldebaran
Aldebaran (Taurus) is a very bright orange-red star. We observe that its peak emission is at 725 nm (red) and that
it is luminosity including IR (the total amount of power it emits) is 425 times more than the Sun.
(2 pts) What is the surface temperature of Aldebaran?
(2 pts) How much power per unit surface area does Aldebaran emit? Answer in W/m2.
(1 pt) LAldebaran = 425 LSun and LSun = 3.85x1026 W. How much total power does Aldebaran emit? Answer in W.
(2 pts) Combining the above, what is the surface area of Aldebaran? Answer in m2.
(2 pts) What is the radius of Aldebaran? Express your answer in m and AU.
(1 pts) RSun = 7x108 m. How many solar radii is Aldebaran? I
(2 pts) The Sun has an angular size of 0.4o. If Aldebaran were our sun (we are 1 AU away), what would it’s angular
size be?

                                      ASTR1030 Fall 2009. Homework 3. Page 7
13. The Moon
The center of the Moon is an average distance of 384,000 km from the center of the Earth (radius of 6376 km).
(5 pts) Use these facts, and look up the mass of the Earth and the Moon, to determine the center of mass between
the Earth and Moon. Is this point above or below the surface of Earth?

                  Center of mass.

14. Cosmic Microwave Background
The diagram to the right displays the cosmic microwave back-
ground (CMB) as measured by the COBE satellite. This back-
ground radiation was predicted by the big bang theory prior to
the COBE measurement. The prediction and the measurement
agree so precisely that they overlay each other. For this achieve-
ment, the scientists were awarded the Nobel prize in physics.
(2 pts) At what wavelength does the CMB peak (mm)? What is
the frequency of the radio waves (GHz)? CAREFUL! The plot is
in units of inverse wavelength (1/cm).
(3 pts) This background is seen in all directions and appears to
be coming form the edge of the universe. In other words, we are
looking back in time almost 14 billion years. What was the tem-
perature of the universe 14 billion years ago? Answer in K.

FOR THIS PROBLEM ONLY: Wien’s law as we learned,
              6            o
T = ( 2.9 × 10 ⁄ λ peak ) ( K nm ) , applies if the plot is made in inten-
sity per unit wavelength. Sorry to confuse you, but Wien’s law for
plots of intensity per unit inverse wavelength (or per unit fre-
quency) is:
                               5.1 × 10 o
                           T = ------------------- ( K nm )
                                  λ fpeak

                                                   ASTR1030 Fall 2009. Homework 3. Page 8

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