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Astrophysics Annex
While the MFT contains "astrophysics" questions, don't think that you cannot answer them without any
classes in the subject! Yes, last year some questions involved stars, but stars were simply a convenient
real-world blackbody for Wein's Law and the Stefan Boltzmann Relation -- actually just
thermodynamics! This section is a quick introduction to the types of "astrophysics" questions which
may be expected and some explanation of some of the exotic units which might be used in the problem.
Angles: Astronomical measurements typically use
angles to report the apparent size or separation
between two objects (as well as to establish a
coordinate system on the sky). The formula for
angular size or separation is:
tan = s/d 
where d is the distance to the object and
s is either a physical size of an object or the
distance separating two objects.
Many of these angles are smaller than a degree and
have time-based meanings as well so follow the
analogy. 1 degree can be split into smaller units
(arcminutes, or ') where 60' = 1 degree. 1' is the
finest angular separation an average eye can
discern. An arcminute is also split into 60
arcseconds (or "). In short: 1o = 60' = 3600".
Distance: Distance is hard to measure precisely in astronomy and in astrophysics and the exotic units
used typically hint at the technique used in the measurement. With the exception of light-year I would
expect to find any of the other units defined on the exam.
1 Astronomical Unit: The distance from scaled to the distance of the Earth to the Sun: 1.496 X 108 km
1 Parsec: The distance from the Sun to a star with an annual parallax angle of 1": 3.09 X 1013 km
1 Light X (where X= a unit of time like year, day, second) distance is from light travel time: Use d=ct,
where t is the appropriate X in seconds
Escape Velocity: An object trapped in a gravitational potential well of a larger primary object has a
negative gravitational potential energy. In order to escape (to infinite distance) where the potential
energy is zero and the kinetic energy must be  0. If the mass of the object is m, the mass of the
primary object is M and the separation between the two is r, the escape velocity (ve) is determined by:
KE  PE  0
½ mve2 
GMm
0
r
ve 
2GM
r
Kepler's Laws: Gravity drives the universe. Planetary motions
are typically described in terms of Kepler's laws:
1) All planets move in elliptical orbits with the Sun at one focus.
2) The line that joins a planet to the sun sweeps out equal areas
of their orbit in equal times. (Closer to the Sun you move more
quickly -- angular momentum conservation; m r v )
3) For objects of planet size or smaller orbiting the Sun, if we
express the period (P) in years and the average distance
(a, semi-major axis) in astronomical units then P2  a3.
How can years squared equal astronomical units cubed? There is
a hidden scaling! Kepler didn't know Newton's laws, just what
he could observe which was periods and astronomical units (both
defined in terms of the Earth's 1 year and 1 AU). Try
approaching Kepler's Third Law by deriving it from Newton.
Consider a planet (mass m) in circular motion at distance R
around the Sun (mass M, M >> m). The only force acting on the
planet is gravity and it creates a centripetal acceleration:
F= ma=
GmM
v2
a=
R 2 but for circular motion,
R so
v2= GM . But the motion is circular so v  circumference  2  R and 4  2 R3  G M P 2 
r
P
period
Okay, but that also means that
4  2 R3
 1 , so for two planets with distances R1and R2 and periods P1
G M P2
3
4  2 R13
4  2 R23
R13 R23
 R2  2 . Choosing planet 1 to be
P

P
and P2,
.
Cancelling,
or

1



2
1
 R1 
G M P12
G M P22
P12 P22
R

the earth, R1 = 1 AU and P1 = 1 year, and P2   2

(1
AU
.
.)


3
2
yr.
Blackbodies: Stars are blackbodies as discussed in section (22). They have a wavelength at which
they emit the greatest intensity of light (given by Wein's law) and they emit a total amount of radiation
consistent with the Stefan Boltzmann relation.
Stellar Spectra: The composition of stars can be determined because the inner core of a star radiates
like a blackbody. The outer cooler parts of the star absorb certain colors of light due to radiative
excitation and the rest of the spectrum makes it to us minus those colors. The exact wavelengths of
light removed are determined using the techniques in section (21). Each element has a unique set of
energy levels, therefore we can tell what elements are present based on their spectral lines.
Stellar Structure and Evolution: Stars are held together by gravity but are prevented from collapsing
by fusion in their cores. This balance of pressures keeps them stable. For the majority of their lives,
stars turn H into He by fusion and these stars are called "main sequence". It is possible to fuse heavier
elements (most stars are massive enough to generate sufficient pressures and temperatures to fuse He
into C late in their lives when they have depleted the H supply in their cores).
Stars are in hydrostatic equilibrium which you will remember from your general physics class.
G M (r )   r 
dP

dr
r2
In this equation, P is pressure (force/area), r is distance from the center, M(r) is the net mass within r or
the center (net mass located inside a sphere of radius r concentric with the , ρ(r) is the density at
distance r and G is the universal gravitational constant.
Distances (actually best expressed as light travel time):
Moon: When you listen to the communications of the Apollo astronauts there is a couple second lag
between question and answer. They are not thinking hard -- the moon is approximately 1 light second
or 300,000 km away and the pause is the travel time of the radio waves.
The Sun is 8 light minutes away. If the Sun suddenly stopped producing energy, it will still take 8
minutes for the Earth to know it. Incidentally the sun is about 1 light second in radius as well.
Pluto is the edge of the planets we accept in the solar system. It would take 4.5 hours for light to reach
Pluto and twice that time to have a round trip. As we explore the solar system with robots, this time lag
has to be taken into account.
The next star is approximately 4 light years away. In the Galaxy stars are typically
1 light year apart so we live in an uncrowded neighborhood. That's good, because interactions with
stars are relatively disruptive to planetary orbits... Also stars are small compared to their separations -a star is a light second across but a light year away from its neighbor.
The our Milky Way Galaxy is approximately 100,000 light years across (We are 26,000 light years
from the center) and contains 100 billion stars. By rough coincidence there are approximately 100
billion galaxies in the universe so that there 10 thousand billion billion stars in the universe.
Galaxies are large compared to their separation. Our galaxy is 100,000 light years across but the next
large galaxy Andromeda is only
2 million light years away. Forget about a round trip conversation, though.
The edge of the observable universe is still debatabled is between 14 and 180
billion light years across. I'd personally bet on the lower side of
that...
Times
-------------------------------------------------------------------------------Age of universe - about 14 billion years old Age of the Sun - about 4.5 billion years old Lifetime of
the Sun as a Hydrogen burning star - about 9 billion years Time for the Moon to go around the Earth 27 days. Time for the Sun to rotate: 27 hours
Time for a Sun-sized pulsar to rotate: 1 millisecond Time for the Earth to go around the Sun - 365.25
days Time for Pluto to go around the Sun - 248 years. Time for the Sun to go around the Galaxy - 200
million years Time when Andromeda collides with the Milky Way - 3 billion years
- The sun is still a hydrogen burning main sequence star!
--------------------------------------------------------------------------------------Alternately try the "Galaxy Song" by Monty Python from the meaning of life (which can be
downloaded in mp3 from http://www.mwscomp.com/sound.html):
Just remember that you're standing on a planet that's evolving And revolving at nine hundred miles an
hour, That's orbiting at nineteen miles a second, so it's reckoned, A sun that is the source of all our
power. The sun and you and me and all the stars that we can see Are moving at a million miles a day.
In an outer spiral arm, at forty thousand miles an hour, Of the galaxy we call the 'Milky Way'.
Our galaxy itself contains a hundred billion stars. It's a hundred thousand light years side to side.
It bulges in the middle, sixteen thousand light years thick, But out by us, it's just three thousand light
years wide. We're thirty thousand light years from galactic central point. We go 'round every two
hundred million years, And our galaxy is only one of millions of billions In this amazing and
expanding universe. The universe itself keeps on expanding and expanding In all of the directions it
can whizz As fast as it can go, at the speed of light, you know, Twelve million miles a minute, and
that's the fastest speed there is.
So remember, when you're feeling very small and insecure, How amazingly unlikely is your birth, And
pray that there's intelligent life somewhere up in space, 'Cause there's bugger all down here on Earth.