|
|
Monster of the Milky Way
|
|
|
|
Classroom Activity
|
Activity Summary
In Part I, students use a balloon and aluminum foil ball model to
explore changes in density vs. volume as a massive star evolves into
a black hole. In Part II, students turn to calculations to discover
the implications of increasing density with decreasing size.
Learning Objectives
Students will be able to:
- glass of water
- piece of wood
- aluminum pellet
- iron rod
- piece of gold jewelry
- overhead transparency
- set of different-colored transparency pens
-
copy of the "Dense, Denser, Densest?" student handout (PDF
or
HTML)
-
copy of the "Getting Really Dense" student handout (PDF
or
HTML)
-
50-cm-long piece of aluminum foil (by standard 30.4 cm wide)
- one 22.86 cm (9-in) balloon
- 44-cm-long piece of string
- pan balance
- tape measure
- graph paper
- calculator
Background
A star spends most of its life fusing hydrogen into helium. During
this time, a balance exists between the energy being released and
the gravity of the star. Once a star uses up all the hydrogen in its
core, gravity takes on a bigger role. Gravity's influence on the
core causes it to contract further, setting off a cascade of fusion
of lighter elements into heavier ones. In very massive stars this
cascade continues until only non-fusible iron remains. During this
process, the core releases a far greater amount of energy, which
radiates outward and expands the gas in the outer layers of the
star. As a result, the star swells. Even though the total amount of
energy emitted goes up, because of its large size, the star actually
cools off. It becomes red and bloated, so astronomers call this kind
of star a "red giant."
Eventually, gravity once again comes to dominate. If the star is
massive enough (a mass more than about 20 times greater than that of
our sun), then it can fuse iron nuclei. Because iron takes more
energy to fuse than it releases, the core is robbed of the heat it
needs to balance gravity. (This process also absorbs electrons,
which help support the core against its own crushing gravity.) When
iron begins to fuse, it's like the legs are kicked out from under
the core. It collapses inward within only a fraction of a second,
releasing a vast amount of energy which flashes outwards, tearing
off the outer layers of the star. The star explodes in an event
called a supernova.
What happens to the core when it collapses depends on its mass. If
it is between 1.4 and three to four times as massive as our sun, it
will become a dense neutron star (a neutron star is about 11
kilometers in diameter; a teaspoonful of it weighs about a billion
tons—as much as all the cars on Earth would weigh). Up to a
certain size, a neutron star can resist the inward pull of gravity.
But if it is more than two and a half solar masses, gravity wins and
the neutron star collapses into a black hole. When this happens, the
core digs itself deep into the fabric of spacetime, crushing the
matter itself right out of existence. All that remains is a region
of extremely curved spacetime, the ghost of the collapsed stellar
core, the center of which is called a singularity.
Because black holes are so compact, their gravitational pull is such
that once something ventures too close, it cannot escape. Not even
light can break out. The point of no return is the event horizon
surrounding the black hole. Matter within the horizon falls
inexorably down to the very center, where it gets crushed in the
black hole's singularity, an unfathomable place of collapsed space
and time, where the known laws of physics break down.
The radius of this sphere-shaped event horizon, known as the
Schwarzschild radius, varies according to the black hole's mass. The
expression that determines the radius of the event horizon is:
R = 2GM/c2
where R is the radius of the event horizon, M is the
mass of the black hole in kg, G is the universal
gravitational constant, and c is the speed of light (G = 6.67
x 10-11m3/kg-sec2 and c = 3 x
108 m/sec)
Material falling into the black hole forms an accretion disk of gas
and dust outside of the event horizon, which spirals inward toward
the black hole. In addition, the black hole may have jets of hot gas
and energy streaming outward perpendicular to the accretion disk.
Some scientists claim that black holes open onto other universes. No
one really knows. While black holes have been studied
mathematically, no one has directly observed one. Astronomers infer
the existence of black holes from the effect they have on the
material around them (i.e., observing X-ray emissions resulting from
gas being heated near a black hole due to its strong gravitational
pull). Our Milky Way galaxy has a supermassive black hole in its
with a mass four million times that of the sun. Moreover, every
decent-sized galaxy likely has one of these supermassive black holes
in its core.
In the first part of this activity, students use aluminum balls to
model the formation of a black hole. Students associate the physical
act of crushing aluminum foil (using mechanical forces) into smaller
and smaller spheres with the gravitational effects on a collapsing
star. In the second part of the activity, students calculate the
aluminum ball's density as it becomes smaller and smaller. They also
consider how small it would need to be to become a black hole.
Key Terms
black hole: A collapsed star that has a region surrounding it
from which nothing can escape, not even light.
density: A characteristic of matter defined by the amount of
mass of material per unit volume.
event horizon: The boundary of a black hole that marks the
region surrounding the black hole from which nothing can escape. The
radius of this region is known as the Schwarzschild radius.
fusion: The process in which the nuclei of atoms combine to
form larger ones at high pressure and temperature.
gravitational force: The force of attraction between objects
that contain either mass or energy.
mass: The amount of matter a sample of material contains,
measured in grams or kilograms.
neutron star: An extremely small, super-dense star formed as
a result of a supernova explosion.
Schwarzschild radius: The radius of the event horizon of a
black hole.
volume: The amount of space matter occupies.
Part I
-
Prior to the activity, set up the balances at student lab
stations. Create enough 50-centimeter sheets of aluminum foil
for each team.
-
To introduce the activity, ask students what they know about
black holes. Ask them how they think black holes form. Use the
background information to review how black holes are formed,
what their components are, and where they are believed to exist.
-
Explain to students that they will be modeling how a star's
interior, called the core, evolves into a black hole. The
balloon students blow up represents a star 20 times the mass of
our sun shining steadily as it converts its interior reserves of
hydrogen to helium, a process known as nuclear fusion. The
tension in the balloon material represents gravity trying to
collapse the star's core, and the pressure from the air inside
the balloon represents the heat in the star's core trying to
make it expand. The aluminum foil represents the star's core.
Guide students through what each stage represents prior to or
during the activity:
-
The first aluminum ball that students form symbolizes the
star's core during most of its lifetime. Ask students to
imagine that this core is inside the balloon and is
surrounded by layers of the star as it burns its hydrogen
fuel. The star itself is vastly larger than the aluminum
ball.
-
The first compaction represents what happens after the star
burns through its hydrogen fuel, becomes a red giant, and
then cools and begins to fuse lighter elements into heavier
ones, all the while becoming increasingly more dense. The
star alternately heats and cools during this time and
gravity takes on a bigger role each time the star cools.
Have students continue to visualize the core inside the
balloon.
-
The second compaction symbolizes the iron core that is left
after the star has burned through all its lighter elements.
When iron fuses, it takes away energy from the core that is
needed to support the star. Once this happens, gravity takes
the upper hand and the star's core collapses under its own
weight. The star goes supernova.
-
Crushing the aluminum ball down further with the hammer
symbolizes the hot, dense core left behind after the star
goes supernova. (Students will pop the balloon prior to
making this compaction to symbolize the supernova
explosion.) At this point, a quantum mechanical effect
called neutron degeneracy pressure prevents further
collapse. However, if the neutron star is more than two and
a half solar masses, gravity is so strong that even the
nuclear forces from the neutron star can't stop it—the
core will collapse and become a black hole.
-
Discuss the limitations of this model with students.
(This model is intended to approximate the stages a star goes
through on its way to becoming a black hole; it does not
accurately represent the sizes of the core or the layers
surrounding a star at various times in its life cycle or of
the actual compaction of matter in a star.)
-
Prior to beginning the activity, discuss density with students.
Show students the glass of water, piece of wood, aluminum
pellet, iron rod, and piece of gold jewelry. Ask students which
items they think are most dense. Why? Discuss density, providing
the density of the objects you have shown:
- wood (.5 g/cm3)
- water (1 g/cm3)
- aluminum (2.7 g/cm3)
- iron (8 g/cm3)
- gold (19.3 g/cm3)
Point out that the average density of the sun is 1.4 g/cm3
(note that the sun is about 1/20th the mass of a star
that would become a black hole). Inform students that as forces
are applied to most (non-liquid) substances and they are
squeezed into a smaller space or volume, their density changes.
-
Draw a one-centimeter cube on the blackboard. Ask students which
of the following elements would weigh the most (be the most
massive) within the box: water, aluminum, or gold.
(Gold, the densest naturally occurring element on Earth. The
box will hold 1 gram of water, 2.7 grams of aluminum, or 19.3
grams of gold.)
-
Organize students into teams and distribute the student handouts
and materials to each team. Review activity directions with
students.
-
Have students inflate their balloons until the string goes just
around the widest part of the balloon (about twice the size of a
grapefruit) and then tie the balloons off. Then have students
use string or measuring tape to measure, in centimeters, the
balloon's diameter in two directions, at 90-degree angles to
each other, to find the average diameter and radius. Help teams
through this process if needed.
-
Have students follow the directions listed on their "Dense,
Denser, Densest?" student handout. After students have
compressed and remeasured their aluminum ball twice they will
bring it to you to hammer down further. Note that it is fairly
difficult to compress the foil to a sphere with a diameter of
less than two centimeters. (See
Activity Answer for a typical value.)
-
Have each team report its data points for you to plot on the
overhead (plot volume on the y-axis and density on the
x-axis). Use a different color for each team's results.
This will allow you to discuss reasons for any data differences
and delete any outliers before you draw a final best-fit curve.
The graph will appear to be headed to zero. Ask students whether
the aluminum ball really goes away.
(In the case of the creation of a black hole, for all
practical purposes the matter collapses to infinite density,
but mathematically it does not disappear entirely. You may
want to note to students that if you were to continue to
magnify the graph, it would come very close to
zero—about 10-50—but would not reach
it.)
-
To conclude, hold up the aluminum pellet again. Tell students
that the aluminum pellet demonstrates the maximum density of
aluminum obtainable on Earth (2.7 g/cm3). (Forces
between the electrons of each atom repel each other more
strongly than any force on Earth that could make the pellet
smaller.) Explain that only in the collapse of massive stars
during supernova events can forces be exerted strongly enough to
overcome the repulsion of the electrons in the atoms. In fact,
when very large stars go supernova, even the forces of repulsion
in the nuclei of atoms are overcome, enabling the mass in the
core of the star to be compressed until it virtually disappears
and a black hole is formed.
-
If you would like, continue to the next part of the activity to
have students work out on paper how much the density of the
aluminum would need to be increased for it to become a black
hole.
-
As an extension, have students create a time line of the theory
of black holes and observational evidence that supports their
existence.
Part II
-
In this part of the activity, students will calculate the
increasing density of their aluminum stars as they get smaller
and smaller. Have students perform the calculations up until
their aluminum stars are calculated to be the size of neutrons
(the average radius of a neutron is equal to 10-15
m).
-
Once they have completed the neutron calculations, inform
students that as dense as their aluminum star would be if it
were the size of a neutron, that is nothing compared to what
happens when a neutron star becomes a black hole—when the
star becomes unimaginably dense in virtually zero volume! Tell
students that this is a very hard model to conceptualize as
there is nothing that exists in our normal experience that it
can be compared to.
-
Have students perform the final calculation of how small the
aluminum star would have to be to become a black hole, and then
how large the black hole's event horizon would be. Discuss with
students whether it would be possible to create a black hole out
of a piece of aluminum foil.
(It is theoretically possible, but would take more energy
than is currently available.)
-
As an extension, have students calculate the Schwarzschild
radius of the black hole in the center of our Milky Way galaxy,
which is thought to have a mass of about 4 million suns (mass of
our sun = 2 x 1030 kilograms)
R = 2GM/c2 = 2 x 6.67 x
10-11m3/kg-sec2 x (4 x 106
suns x 2 x 1030 kg/sun)/( 3x 108
m/sec)2 = 1.2 x 1010 m or 1.2 x 107
km or 7.4 million miles in radius
Sample Data Table and Graph*
Part I
|
Trial #
|
Mass (g)
|
R1 (cm)
|
R2 (cm)
|
Ravg (cm)
|
V (cm3)
|
D (g/cm3)
|
|
1
|
14.1 |
7.6 |
6.4 |
7.0 |
1,432 |
0.0098 |
|
2
|
14.1 |
4.5 |
4.7 |
4.6 |
406 |
0.035 |
|
3
|
14.1 |
2.5 |
2.3 |
2.4 |
58 |
0.24 |
|
4
|
14.1 |
1.7 |
1.5 |
1.6 |
17 |
0.83 |
* Measurements based on two sheets of foil about 50 centimeters in
length and a balloon about 14 centimeters in diameter. The total
mass of aluminum is 14.1 grams in the example.
Part II
Going down to ... one centimeter
V = 4/3πR3 = 1.33 x 3.14 x (1 cm)3 =
4.19 cm3
D = M/V = 14.1 g/4.19 cm3 = 3.437 g/cm3
And farther down to ... one millimeter
V = 4/3πR3 = 1.33 x 3.14 x (.1 cm)3 =.0019 cm3
D = M/V = 14.1 g/.0042 cm3 = 3370g/cm3
And farther down to ... a neutron
V = 4/3πR3 = 1.33 x 3.14 x (1 x 10-13 cm)3 = 4.0 x 10-39 cm3
D = M/V = 14.1 g/4.2 x 10-39 cm3 = 3.4 x
1039 g/cm3
And finally down to a ... black hole
V = 4/3πR3 = 1.33 x 3.14 x (approaches 0)3 = 0 cm3
D = M/V = 14.1 g/0 cm3 = [infinity]
Your black hole's event horizon
R = 2GM/c2 = 2 x
6.67x10-11m3/kg-sec2 x .0141
kg/(3x108 m/sec)2
= 2.09x 10-29m
The size of the event horizon if our sun were to become a black
hole
R = 2GM/c2 = 2 x
6.67x10-11m3/kg-sec2 x 2 x 1030
kg/(3x108 m/sec)2
= 3,000 m (about 3 km)
What would life be like on Earth if the sun was replaced by a
black hole with the mass of the sun?
There would be no light or warmth because no light could escape the
black hole. Anything falling into the black hole would make X-rays
and possibly gamma rays, sending out lethal amounts of radiation.
Nothing could survive. However, assuming that the black hole was the
same mass as the sun, the Earth would continue to orbit as it does
now.
Would Earth be sucked into the black hole?
Earth would not be sucked into the black hole, assuming the black
hole had the same mass as the sun. Since the mass at the center of
the solar system remains the same, Earth's orbital velocity and path
would not change.
Web Sites
NOVA—Monster of the Milky Way
www.pbs.org/nova/blackhole
Watch the program online, learn about how black holes form, discover
what little black holes are, look at some of the other marvels of
the universe, and more.
Black Hole Encyclopedia
hubblesite.org/explore_astronomy/black_holes/encyclopedia.html
Explains the nature, formation, and existence of black holes.
Black Holes
glast.sonoma.edu/teachers/blackholes/index.html
Provides information about black holes; an educator's guide of
activities; and links to additional lessons, games, and resources
about black holes.
No Escape: The Truth about Black Holes
amazing-space.stsci.edu/resources/explorations/
blackholes/teacher/sciencebackground.html
Contains background information about black holes, including facts
about how they form, how light can be trapped in them, the kinds of
black holes that exist, and more.
Books
A Nature Company Guide: Skywatching
by David H. Levy. Time-Life Books, 1995.
Provides a general overview and discussion of astronomical objects,
including black holes.
The Young Oxford Book of Astronomy
by Jacqueline & Simon Mitton. Oxford University Press, 1995.
Explains many concepts in astronomy from the solar system, galaxies,
and the universe, including black holes.
The "Dense, Denser, Densest?" activity aligns with the following
National Science Education Standards (see
books.nap.edu/html/nses).
Grades 5-8
Science Standard B
Physical Science
Properties and changes of properties of matter
Motions and forces
Grades 9-12
Science Standard B
Physical Science
Conservation of energy and increase in disorder
Structure of atoms
Motions and forces
Science Standard D
Earth and Space Science
The origin and evolution of the universe
Classroom Activity Author
Jeff Lockwood taught high school astronomy, physics, and Earth
science for 28 years. He has authored numerous curriculum projects
and has provided instruction on curriculum development and science
teaching methods for more than a decade.
|
|
How Black Holes Form
Find a description and images related to black hole formation
in this NOVA
slide show.
|
|
