These lecture notes were prepared for Rutgers Physics 341/342: Principles of Astrophysics
by Prof. Chuck Keeton, and modified by Profs. Saurabh Jha and Eric Gawiser. All rights
reserved.
c
2011
Lecture 6: Supermassive Black Holes in Other Galaxies
I.
Technical Background: Doppler Effect
(See
§
4.3 of Carroll & Ostlie for more details.)
You are probably familiar with the Doppler effect: when a train/airplane/racecar passes by,
the sound shifts from a high pitch to a low pitch. In 1842, Christian Doppler explained this
effect by noting that as a source of sound moves toward you, the sound waves get compressed
so the frequency (pitch) increases; while when the source moves away, the waves get stretched
out so the frequency goes down. He showed that the change in wavelength can be written as
λ
obs

λ
rest
λ
rest
=
v
c
s
where
c
s
is the speed of sound, and
v
is the speed with which the source is moving away
from you (so
v <
0 if the source is moving toward you).
Figure 1: (From http://odin.physastro.mnsu.edu/˜eskridge/astr101/kauf5
23.JPG.)
The same thing happens for light, as shown above, but the details are slightly different
because of relativity. Specifically, when we take relativity into account (as we will see later
1
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in the semester), we arrive at the exact
relativistic Doppler formula
:
λ
obs
λ
rest
=
s
1 +
v/c
1

v/c
When the speed of the source is slowed compared with the speed of light, we can use our
favorite Taylor series approximation
1
to write:
λ
obs
λ
rest
=
1 +
v
c
1
/
2
1

v
c

1
/
2
=
1 +
1
2
v
c
+
O
v
c
2
1 +
1
2
v
c
+
O
v
c
2
=
1 +
1
2
v
c
+
1
2
v
c
+
O
v
c
2
=
1 +
v
c
+
O
v
c
2
So when
v
c
, we derive the
nonrelativistic Doppler formula
λ
obs
λ
rest
≈
1 +
v
c
We sometimes write this in terms of the
redshift
z
, defined by
z
≡
λ
obs

λ
rest
λ
rest
Given this definition, note that
z
=
λ
obs
/λ
rest

1, and in the nonrelativistic case
z
≈
v/c
.
Now the idea is that if we
measure
the observed wavelength
λ
obs
of some light, and we
know
the corresponding rest wavelength
λ
rest
(e.g., from lab measurements), then we can use these
formulas to determine the speed with which an object is moving toward or away from us.
II.
NGC 4258 = M106
After the Milky Way, the galaxy where we have the best constraints on a supermassive
black hole is NGC 4258. Radio observations reveal that there are
water masers
moving in
circular orbits around the center of the galaxy. Maser stands for microwave (or molecular)
amplification by stimulated emission of radiation; it is basically the same thing as laser
(which after all is an acronym for light amplification by stimulated emission of radiation). A
maser emits light at very specific wavelengths, so we can use the Doppler effect to measure
the velocity.
1
As always, (1
±
x
)
α
= 1
±
αx
+
O
(
x
)
2
. So if
x
1, then (1
±
x
)
α
≈
1
±
αx
.
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 Fall '11
 Gawiser
 Black Holes, Mass, Black hole, supermassive black hole, Quasar

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