Download Homework 1 - Concord University

PHYS 420: Astrophysics & Cosmology
Homework 01
Due in class: Thursday, Jan 26
1. A Gut Feeling for Timescales [5 points]. We will see that the age of the universe
is about 14 billion years = 1.4 × 1010 yr. Such enormous timescales are hard to
grasp in an intuitive way. To begin to develop a feel at least for the large numbers,
we will scale the problem down in considering large numbers of seconds rather
than years.
a. Consider the time span of 1 million seconds = 106 sec. Rewrite this in a
more convenient familiar unit–i.e., hours, days, months, years, or centuries.
Will a semester of PHYS420 take more or less than a million seconds? (In
real time–not how it seems!)
b. How many million seconds are in a year?
c. Now consider 1 billion seconds = 109 sec. Do you know anybody this old?
What historically was going on 109 sec ago?
d. If the universe were 14 billion seconds old, how old would that be, in more
familiar units (years, centuries)? Replacing each second with a year, you
perhaps begin to get an idea of the tremendous (but finite) age of the
universe.
2. Telescopes as Time Machines. It is crucial for astronomy and especially
cosmology that the speed of light, c, is finite (c=3.0 × 108 m/s) Because of this,
telescopes are time machines. Indeed, even your naked eye is a time machine.
a. [5 points]. Estimate the length, in meters, of the PHYS420 classroom
(Science Hall 309). Then compute the time it takes for light to travel from
the front to the back of the room. About how far back in the past is the
lecture, as seen by someone seated in the front row? the middle row? the
back row? Comment on why these different delays don’t make for
enormous confusion.
b. [5 points]. The Moon orbits the Earth at a radius of 360,000 km. How long
does light take to go from the Moon to the Earth? Comment on how this
delay figures into the radio transmissions with lunar astronauts. (If you are
curious to test your answer, audio for these can be found online in various
NASA sites!)
c. [5 points]. Now compute the time delay to Mars, when it is at its closest and
most distant distances from the Earth (note that aMars = 1.4 AU). Comment
on implications for the Mars rovers and for future Mars astronauts.
PHYS 420: Astrophysics & Cosmology
Homework 01
Due in class: Thursday, Jan 26
d. [5 points]. Find the light travel time to the nearest star, α Centauri, located
at 1.3 parsec (1 parsec = 3.1×1018 cm). Imagine there are space aliens on
αCen, then (i) sketch one, and (ii) comment on what they see going on here
when they look at us with high-power telescopes and/or tune in to our TV
transmissions (which leave Earth as radio waves).
e. [5 points]. The nearest galaxy like our own is the Andromeda galaxy
(nickname: M31), which is 0.7 Mpc = 0.7 × 106 parsecs away. What would a
space alien in M31 see if they looked today at the Earth with a highpowered telescope?
f. [5 points]. Explain how we can uncover (most of) the past history of the
universe by looking out across the cosmos.
3. Atoms in the Observable Universe [5 points]. Most atoms in the universe are in
the form of hydrogen. On average, hydrogen atoms in the universe today are
found to have a number density (number of particles per unit volume, sometimes
also called a “concentration”) of 2 × 10−8 atoms/cm3. We will later see that the
observable universe today has radius of about R = 4000 Mpc, where 1 Mpc = 1
megaparsec = 106 parsecs, and 1 parsec = 3.1×1018 cm. Find the volume of the
observable universe today, in cm3. Then find the number of hydrogen atoms in the
observable universe today, compare this to 1 googol (that is, the number 10100, not
the search engine!), and comment.
4. The Motion and Layout of the Solar System [10 points].
For the more technically adventurous:
The Greek astronomer Aristarchus
proposed an ingenious method for
determining the relative scale of the
Earth-Moon orbit relative to the EarthSun orbit. Consider the diagram at right.
a. The first quarter moon is the phase
when exactly ½ of the moon is
illuminated–the right-hand side.
The third quarter moon is the phase when exactly ½ of the moon is
illuminate–but now the left-hand side. Explain why this means that the
angle β = 90◦, and thus the upper and lower triangles are similar and are
right triangles.
PHYS 420: Astrophysics & Cosmology
Homework 01
Due in class: Thursday, Jan 26
b. A bit of thought will show that the angle α < 90◦. It turns out that the best
way to measure α is to measure the two time intervals, t1Q→3Q from first to
third quarter moons, and t3Q→1Q from third to first quarter moons.
Assuming the Moon moves in a circular orbit at constant speed, find an
expression for α (in degrees or in radians) given these two time intervals.
Hint: the moon’s period for one complete orbit must be P = t1Q→3Q +
t3Q→1Q.
c. Now that we know how to get α, find an expression that uses only α to find
the ratio r/a, where r is the Earth-Moon orbit radius and a = 1 AU is the
Earth-Sun orbit radius. This then gives the Moon’s orbit radius in AU, so
that we can draw the above diagram to scale (which it is not!).
d. For style points: Let’s turning the problem around: find the actual values of
r and a. Use these to calculate the value for α. How difficult will this be to
measure based on naked eye observations?
5. Universal Gravity. With Newton’s theory of gravitation, we can explain the
motion of ALL objects in “free-fall”, whether on the surface of the Earth or in a
galaxy far, far away.
a. [5 points] By dropping ordinary household objects in ordinary households
(and laboratories) we find the acceleration due to gravity at the Earth’s
surface is g = 9.8 m/s2. Use this result, the radius of the earth R = 6400 km
= 6.4 × 106 m, and the laboratory value G = 6.7 × 10−11 𝑚" 𝑘𝑔𝑠 ' to
calculate the mass of the Earth.
b. [5 points] Compile a table of planetary semi-major axis and masses, ordering
the table with increasing semi-major axis. In a 3rd column, calculate the
enclosed planetary mass as a function of the distance from the Sun. How
does the final cumulative planetary mass compare to the solar mass (𝑀⨀ =
1.989×10"1 𝑘𝑔)?
c. [5 points] In galaxies, including our own, we can determine the orbit speeds
of stars at various distances from the galaxy center. If we know the speed at
a particular distance, this gives a measure of the mass interior to that
distance (i.e., the mass “enclosed” by the star’s orbit). Explain how we can
use data on orbits at different distances to determine the mass distribution
of a galaxy.
PHYS 420: Astrophysics & Cosmology
Homework 01
Due in class: Thursday, Jan 26
d. Alternative Universes: A New Gravity Law [10 points]. In class we saw that
Newtonian gravity provides an incredibly successful theory of motion in the
solar system and in many other astronomical systems To get an appreciation
for how Newton’s gravity affects our lives, let’s imagine a universe with a
different gravity law. In this “bizarro” universe, the gravity force between
two masses (call them m1 and m2) is still attractive, is still directed along a
line between them, and is still proportional to each of the masses, and still
depends on the distance R between the two masses.
In our real universe, Newton’s gravity force depends on the inverse square
of the distance, so we have a gravity force Fgrav,us = Gusm1m2/R2. However,
in the alternate bizarro universe, the gravity force depends directly on the
distance, so that the gravity force is Fgrav,bizarre = Gbizarrem1m2R, where Gbizarre
is a constant number measured by the bizarro universe cosmologists. What
are some ways in which the bizarro universe be different from ours? Do not
do any calculations, but think about what would happen as a result of the
difference in the gravity force law. Hint: you might think about what (if
anything) be different for planets, the Sun, the solar system, or our Galaxy).