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Santa Monica College *
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Subject
Astronomy
Date
Apr 26, 2024
Type
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5
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Black Holes
INTRODUCTION
The existence of black holes was originally predicted by Einstein’s Theory of General Relativity
over one hundred years ago. But without any observational evidence of their existence, they
remained merely a theoretical possibility for many decades. Today, we know without a doubt
that black holes are real, physical objects in space. Indeed, we have numerous independent
observations that point conclusively to their existence.
One of the most compelling sources of evidence for black holes is shown in the animated loop
below. Since 1995, astronomers from UCLA have been observing a group of stars near the very
center of our Milky Way galaxy - a region where astronomers predict that there is a
supermassive black hole. While the black hole itself is invisible, its gravitational effect on nearby
stars can be observed.
In this lab, we’re going to try to make sense of this data and use it to determine the size of the
black hole at the center of our galaxy.
UCLA’S GALACTIC CENTER DATA
The animation above looks complicated (because it is), but let’s walk through it step by step.
●
Watch this
video of UCLA astronomer and Nobel Laureate, Andrea Ghez
, describing how
the motion of the stars is recorded by taking a series of photographs through a
telescope.
●
Notice in the upper right-hand corner of the animation that the date of each observation
is given. This data set was captured between 1995 and 2012.
●
The path of the stars over time is added onto the animation so that we can more easily
track the orbit of each star.
HOW BIG IS THE BLACK HOLE?
When discussing the “size” of a black hole, we can talk about two different properties:
1)
the
mass
(M) of the black hole, as measured in solar masses, or
2)
the
Schwarzschild Radius
(R
sh
), which is the area around the black hole from which
nothing can escape, not even light. We usually measure R
sh
in units of kilometers.
Fortunately, both properties are related to each other through the equation for the
Schwartzschild Radius.
R
sh
= 3·M
So, if we can find the mass of the black hole in solar masses, then we can simply multiply by
three to find its Schwarzschild Radius in kilometers.
CALCULATING MASS
To calculate the mass of the black hole, we need to examine the orbital motion of the stars
around it. In particular, we need to determine the orbital period (P) and the semi-major axis (a)
of the orbits.
The orbital period (P) is the time (in years) that it takes an object to complete one orbit. The
semi-major axis (a) is
one half
of the major axis of the elliptical orbit.
With that information, we can use the simplified version of Kepler’s 3rd Law to calculate the
mass.
M = a
3
/ P
2
So now the method is clear. If we can measure the orbital period and the semi-major axis for
just
one
star orbiting the black hole, then we can calculate the black hole’s mass and
Schwarzschild Radius. If we can perform this measurement and calculation for
two stars
orbiting
the black hole, then we can even double-check our values. That’s what we will do!
DATA COLLECTION
We will focus our data collection efforts on just two of the stars in this field, SO-2 and SO-102.
First, let’s try to determine the
orbital period
of these two stars.
Use this
video of the galactic center
to determine how many years it took each of these two
stars to complete one full orbit (
please note that the measurements for the star S-102 does
not begin until 7 seconds into the video
). Use the pause button and the player bar to move
carefully back and forth through the images, trying to obtain the most accurate measurement
that you can. Record each star’s orbital period in the table below.
Next, we will try to determine the
semi-major axis
of these two orbits in astronomical units
(AU). This
still image from the data set
shows that the observations are calibrated with an arrow
indicating an angular size of 0.1” (that is, a tenth of an arcsecond). At the galactic center, this
arrow corresponds to a physical size of 812 AU. Use this information, along with a ruler to
measure the semi-major axis of each orbit as accurately as you can. (It may help to print the
still
image with inverted colors
, so you can measure on a flat hard surface.)
Remember that the semi-major axis is
one half
of the major axis of an elliptical orbit. (This step
involves a little tricky math. If you need help,
watch this video
.)
Enter your values in the table below
.
Star
Orbital Period (P)
in Years
Semi-Major Axis (a)
in AU
SO-2
16 years
726.512 AU
SO-102
12 years
747.88 AU
CALCULATIONS
Use the simplified version of Kepler’s Third Law, along with the data you collected, to calculate
the
mass
of the black hole.
Enter your values in the table below
.
Measurement
Black Hole Mass
in Solar Masses
Schwarzschild Radius
in Kilometers
Based on SO-2 Data
1,255,543
3,766,629
Based on SO-102 Data
2,423,495
7,270,485
The two stars provide two independent measures of the mass. The difference between these
two measurements gives us a sense of how accurate our values are. If you notice a large
difference, don’t be discouraged - remember that we did all of this data collection with a ruler
and a video! Astronomers have much more sophisticated methods of analyzing the data.
Even so, getting two independent measurements that have the same place value is reassuring.
Based on your data and analysis, you should be able to confidently answer the question...
The black hole at the center of our galaxy has a mass that is equivalent to
(
Circle, underline or make bold your answer below
)
a.
hundreds of suns
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