Lab 9 - Mike Jacobs
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Astronomy
Date
Dec 6, 2023
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10
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1
PHYS 1403 Lab 9: BLACK HOLES
Worksheet
Name:
_________________________________________________________
CWID:
_________________________________________________________
INTRODUCTION
Although the lifetimes of stars are very long, all stars will eventually die. There
are three possible end results of a star. If the star is smaller than approximately 1.4
solar masses, the star will shed its outer layers to form a shell of gas and dust called a
planetary nebula. This will leave the star's core behind as an extremely dense,
extremely hot remnant, called a white dwarf. White dwarfs are about the size of the
Earth, and a tablespoon of white dwarf matter would weigh several tens of thousands of
tons, about as much as an ocean liner.
If the star is larger than 1.4 solar masses, but smaller than about 3 solar masses,
it will go through a series of contractions and expansions. The death of the star will
occur when the nuclear fires in the core no longer generate enough heat to balance the
gravitational attractive forces of the star's mass, leading to a collapse of the mass into a
super-dense, compact core. During this final collapse an enormous amount of energy is
stored within the star's nuclear structure. When this stored energy is high enough to
stop the collapse of the star, it will suddenly be released in a spectacular supernova
explosion brighter than an entire galaxy. A super-dense core, called a neutron star, will
remain where the star once was. Although called stars, neutron stars are not normal
stars because there is no fusion taking place within them anymore. Neutron stars have
a radius of a few tens of kilometers and a tablespoon of neutron star matter would
weigh billions of tons.
Initially, both white dwarfs and neutron stars are very bright, due to the extreme
surface temperature. But they will gradually cool down over a period of millions of years
until they become cold, dark embers. It is speculated that there are countless numbers
of these objects traveling through space, too dark for us to observe them.
If the star is larger than approximately three solar masses, it will go through the
same contraction and expansion as the star that went supernova. But when it goes
through the final collapse, the gravitational forces are so strong that they overcome the
energy stored in the nuclear structure. This star will continue to collapse upon itself until
the entire mass of the star is concentrated at a single point, called a singularity. It has
2
now become so dense that nothing, not even light, will be able to escape from it. Since
anything caught in its gravitational field cannot ever hope to escape, it can be thought of
as a 'hole' in space. And since no light is emitted by it, it is a 'black hole'. This term was
first used by the American physicist John Archibald Wheeler in the 1960's.
Let's see how it is that not even light can escape from a black hole. An object on
a planet's surface has a potential energy due to the planet's gravity. This potential
energy can be found with the equation
R
GMm
U
=
where
M
is the mass of the planet,
m
is the mass of the object, and
R
is the radius of the
planet.
G
(= 6.67 x 10
-11
Nm
2
/kg
2
) is the universal constant for gravity discovered by Sir
Isaac Newton.
If we were to fire this object upward from a planet's surface and make it a
projectile, it would rise up with some starting velocity, but would immediately begin to
slow down until it momentarily came to rest in mid-air, and then return to the surface.
This is something you've experienced before whenever you have thrown a ball into the
air. As you know, if we make the starting velocity larger, that is, if we throw the ball
harder, the projectile will go higher in the air. The larger we make the starting velocity,
the higher the projectile will go, until eventually, we can make the starting velocity so
large the projectile will go up and never fall down. In other words, we have now placed
our object in orbit. If we were to give it a little more energy it would then be able to leave
the planet's gravitational field and never return. This velocity is called the 'escape
velocity'.
The energy from the projectile’s velocity, its kinetic energy found with the
equation
2
2
1
mv
K
=
which must be large enough to overcome the potential energy. So, we have
R
GMm
mv
esc
>
2
2
1
3
Solving to find the escape velocity gives us
(1)
R
GM
v
esc
2
>
Note that the escape velocity does not depend on the mass of the projectile, but it does
depend on one over the radius of the planet. This means that as the planet gets smaller,
with its mass staying the same, the escape velocity will get larger. Well, we saw that
neutron stars and black holes are the result of stars collapsing. This means that the
mass has remained large, but the radius has decreased. By our equation (1) above, we
would expect the escape velocity of these objects to increase as they collapse. A black
hole is a collapsed star whose
v
esc
is greater than the speed of light, which is equal to 3
x 10
8
m/s, and which we denote with the symbol '
c
'.
Using equation (1), and substituting
the value of the speed of light for
v
esc
, we can then solve for the radius that would be
required to have an escape velocity equal to the speed of light.
This result is
(2)
2
2
c
MG
R
SCH
=
What we have found here is the minimum radius that a given mass must
compress to make its escape velocity equal to the speed of light. This radius was first
solved for in 1916 by the German physicist, Karl Schwartzschild, and is known as the
Schwartzschild radius. As a collapsing star passes its Schwartzschild radius, it becomes
a black hole. This is the radius a black hole would have if it were not rotating. The core
would continue to collapse further, but we would not be able to see anything within the
black hole.
The Schwartzschild radius marks what is known as the 'event horizon', beyond
which we can have no knowledge of events that may occur. In other words, if we were
close enough to see a black hole, what we would see is the event horizon. The actual
star, and anything else that had passed through the event horizon, would be inside the
event horizon and will never be seen again.
The star remnant becomes a black hole when it reaches this point because of
two principles described by Einstein in his Theory of Relativity, (1) nothing can go faster
than the speed of light; and (2) light is attracted by gravity. This explains why we call
them 'black holes'. The gravitational field is so strong that nothing, not even light, can
reach the escape velocity. The light is still emitted by the singularity within, but its
trajectory is bent until it falls back onto the singularity without ever passing through the
event horizon.
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It is important to note that the Schwartzschild radius is valid for a non-rotating
star only. Since we believe all stars rotate, their singularities would also rotate and the
true radius of the event horizon would be different. However, this calculation was so
complex that it defied all attempts to conduct it. Finally, in 1963, the Australian
mathematician Roy P. Kerr was conducting some calculations on another aspect of
relativity when he realized he had calculated the radius of a rotating black hole. Such a
black hole is now called a Kerr black hole. However, we will concern ourselves with
Schwartzschild black holes only.
Next, imagine an object emitting light photons as it approaches a black hole. As
this object gets closer and closer to the black hole, but is still outside the event horizon.
As it gets continues to get closer, the trajectories of the light photons are bent more and
more and
more and more of the photons would enter the event horizon. Some of the
photons would be radiated away at angles that would allow them to escape from the
black hole, even with their trajectories bent towards it. However, some would be
radiated at just the right orbit to neither escape, nor pass into the event horizon. These
photons would go into orbit around the black hole and form what is called the 'photon
sphere'. We can find the radius of the photon sphere via relativity to obtain:
(3)
2
3
c
MG
R
PS
=
This means that a black hole has a sphere of photons orbiting it at this radius and
anything approaching a black hole must pass through this sphere before entering the
event horizon.
5
PROCEDURE
Enter you answers to each question in the data tables and indicated spaces below (this
document is form-fillable). When completed, please upload this file using the lab
submission link in Canvas (please submit this file in .pdf format only). Please be sure
that
all
questions are completed.
PRELAB QUESTIONS
1. List the three final forms of dead stars.
2. What is a planetary nebula?
3. What is a supernova?
4. How massive must a star be to become a black hole?
5. What is the Schwartzschild radius?
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6. What is the escape velocity from within a black hole?
7. What is the photon sphere?
8. What are the two principles from the Theory of Relativity that make black holes
possible?
9. What is a Kerr black hole?
10. Where did the term 'black hole' come from?
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EXERCISE
1.
To become a black hole, an object must have a mass greater than three times
the mass of the sun. This is because the forces attributed to quantum mechanics
will keep anything smaller from becoming dense enough.
But we can calculate
what the Schwarzschild radius of an object would be as if it could become a
black hole. Using equation (2) from Lab 9 instructions, calculate the
Schwarzschild radius of the following objects:
a) A 60 kg person.
R
sch
= ________________________________________________________
b) The Earth (6 x 10
24
kg)
R
sch
= ________________________________________________________
c) Jupiter (2 x 10
27
kg)
R
sch
= ________________________________________________________
d) The Sun (2 x 10
30
kg)
R
sch
= ________________________________________________________
e) A star with three solar masses.
R
sch
= ________________________________________________________
2.
Use equation (3) from the Lab 9 instructions to calculate the radius of the photon
sphere for each of the objects in question 1.
a) A 60 kg person.
R
ps
= ________________________________________________________
8
b)
The Earth (6 x 10
24
kg)
R
ps
= ________________________________________________________
c)
Jupiter (2 x 10
27
kg)
R
ps
= ________________________________________________________
d)
The Sun (2 x 10
30
kg)
R
ps
= ________________________________________________________
e)
A star with three solar masses.
R
ps
= ________________________________________________________
3.
The concept of a black hole depends on the speed of light being the universal
speed limit. But why can't something go faster than this? Imagine a car going 100
km/hr firing a bullet going 1000 km/hr, a witness standing by the road would see
the bullet traveling at 1100 km/hr. This final velocity is simply the sum of the two
velocities (100 km/hr + 1000 km/hr). So why can't an object within the event
horizon be traveling very fast (say 1/2 the speed of light, .5
c
) and be emitting
light? Wouldn't the light emitted then be traveling at 1.5
c
? This question was
answered by Albert Einstein in 1905 with his Special Theory of Relativity. The
answer involves the concept that space and time are inter-related and
dimensions and time is effected by the velocity of the object. In other words, the
faster you go, the more time slows down. So you would be going faster, but your
time would slow down to the point that the velocity of your emitted light will still be
equal to the speed of light. This effect is shown in the Lorentz factor, designated
with the Greek letter gamma (
γ
), where
(5)
2
1
1
−
=
c
v
γ
This factor shows how much distance, time and velocity are effected by motion
9
relative to an observer or some 'fixed' reference point. In this way Einstein was
able to show all measurements in space and time are relative to the observer,
hence the name of the theory.
This effect is not obvious to us because we travel so slowly relative to the speed
of light. To illustrate this, calculate the gamma factor for the following velocities.
Keeping the velocities in terms of
c
will simplify the calculations.
a) 28 m/s (about 9.33
×
10
-8
c
)
γ
= ________________________________________________________
b) 0.1
c
γ
= ________________________________________________________
c) 0.3
c
γ
= ________________________________________________________
d) 0.5
c
γ
= ________________________________________________________
e) 0.7
c
γ
= ________________________________________________________
f) 0.9
c
γ
= ________________________________________________________
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4. Does a black hole have to be indescribably dense? Would it be possible for us to
approach and enter a black hole without becoming aware of it until it was too
late? To answer this, let's rewrite equation (2) as
G
Rv
M
2
2
=
a) If the radius of the universe is 15 billion light years, how much mass would be
required to make our universe a black hole to an observer 'outside' of our universe?
Remember to convert light years to meters. The speed of light is approximately 3x10
8
m/s.
Mass = _______________________________________________________
b) If the average galaxy has 1 trillion (10
12
) solar masses in it, how many galaxies would
it take to make up the mass you calculated in question 4a?
Galaxies = _____________________________________________________
c) It is estimated that there are approximately 100 billion galaxies in the universe. Based
on this, is possible that the universe is a black hole to an observer 'outside' of our
universe?
Answer = _______________________________________________________