Astronomy 3 Lab Manual
Radio Observing
1
The Mass of the Milky Way
Introduction: The 21-cm line of neutral hydrogen
Hydrogen is the most abundant element in the cosmos; it makes up 75% of the universe’s mass.
Therefore, it is no surprise that one of the most significant spectral lines in radio astronomy is the
21-cm hydrogen line. In interstellar space, gas is extremely cold. Therefore, hydrogen atoms in
the interstellar medium are at such low temperatures ( 100 K) that they are in the ground electronic
state. This means that the electron is as close to the nucleus as it can get, and it has the lowest
allowed energy. Radio spectral lines arise from changes between one energy level to another.
A neutral hydrogen atom consists of one proton and one electron, in orbit around the nucleus.
Both the proton and the electron spin about their individual axes, but they do not spin in just one
direction. They can spin in the same direction (parallel) or in opposite directions (anti-parallel).
The energy carried by the atom in the parallel spin is greater than the energy it has in the antiparallel spin. Therefore, when the spin state flips from parallel to anti parallel, energy (in the form
of a low energy photon) is emitted at a radio wavelength of 21-cm. This 21-cm radio spectral line
corresponds to a frequency of 1420 MHz.
The 21-cm hydrogen radiation is not impeded by interstellar dust. Optical observations of the
Galaxy are limited due to the interstellar dust, which does not allow the penetration of light waves.
However, this problem does not arise when making radio measurements of atomic hydrogen. Radiation from this region can be detected anywhere in our Galaxy.
Areas which contain cold hydrogen gas are called HI regions. Observations of the 21-cm line from
HI regions in our Galaxy can be used to measure the speed of rotation of objects about the center
of the Milky Way. Gravity is holding the Milky Way together and preventing stars and HI regions
from flying off into intergalactic space. The strength of the gravitational force depends on the mass
of the galaxy – more massive galaxies have strong gravitational forces and higher rotation speeds.
As discussed in Mathematical Insight 19.1 of your textbook, measurements of rotation velocities
of objects can be used to estimate the mass of the Milky Way.
In this lab, you will use one of the Student Radio Telescope (SRT) on the roof of Wilder Lab to
measure the velocity of HI gas in the disk of the Milky Way. These telescopes have a diameter
of 2.1m and have a sensitive radio receiver at the focus of the telescope. The radio receiver is
designed to scan a range of wavelengths centered about the 21-cm line of hydrogen. As the gas in
the Milky Way is moving, it is Doppler shifted and the receiver can determine the velocity of the
gas by comparing the observed wavelength to the standard wavelength (21-cm). The telescope and
receiver are controlled remotely, using the computers in the astronomy lab (room 200 Wilder).
From your measurements of the velocity of the gas orbiting the Milky Way, you will be able to
determine the rotation curve of the Milky Way and finally, estimate the mass of the Milky Way.
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Figure 1: Gas clouds rotate around the center of the Milky Way. Clouds at different distances
have different velocities and therefore give rise to emission lines with different Doppler shifts. The
observed flux profile (solid line in figure on the right) is the sum of the line profiles of all the
individual line profiles (dashed lines). The numbers of the line profiles correspond to the clouds in
the picture on the left.
Theory
To first order, we can assume that the gas in the Galaxy is in a circular orbit about the center of
the Milky Way. As discussed Mathematical Insight 19.1 of your textbook, the velocity V of a gas
cloud at a radius R from the center of the galaxy is
r
GMR
Vgas =
(1)
R
where MR is the mass of the Milky Way (enclosed within the radius R) and G is the Gravitational
Constant, G = 6.67 × 10−11 m3 /kg/s2 . Thus, gas clouds at different distances from the center of
the Milky Way, will have different velocities.
As illustrated in Figure 1, when we point our radio telescope in the disk of the Milky Way, we
receive a signal (flux) from a number of clouds along the line of sight. These clouds are orbiting
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Radio Observing
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the center of the Milky Way at different distances and have different velocities. This leads to
different Doppler shifts for each gas cloud. Recall that the Doppler shift (Section 5.5, in your
textbook) only measures one component of the velocity – the velocity along the line of sight (often
referred to as the radial velocity). The maximum Doppler shift (corresponding to the maximum
velocity along our line of sight) occurs for cloud 3 in Figure 1, as for this cloud the circular velocity
is lined up with our line of sight. The other clouds in Figure 1 only have a small fraction of their
total circular velocity lined up with our line of sight, so that the Doppler shift is much smaller. As
illustrated in Figure 1, the maximum velocity (cloud 3) occurs at the tangent point, the point where
the direction of observation is at right angles to the the Galactic Center. This right angle triangle
allows us to easily determine the distance the maximum velocity,
R = Ro sin `
(2)
where Ro is the distance between the Sun and the Galactic Center and ` is the angle that our
telescope is pointing with respect to the line between the Sun and the Galactic Center. This is called
the Galactic longitude. From other measurements, astronomers have determined the distance to the
Galactic center to be Ro = 8 kpc = 8000 parsecs.
The Doppler shift (see Mathematical Insight 5.3 for more details) measures the relative velocity of
the gas cloud along the line of sight with respect to the our (the Sun’s) motion. Thus, the velocity
which is observed by the radio telescope is
Vmax,observed = Vgas −Vsun sin `
(3)
as illustrated in Figure 1. The velocity of the Sun may be written as
Vsun = ωo Ro
(4)
where ωo is the angular velocity of the Sun about the Galactic center. Combining equations equations (3) and (4) we see that
Vgas = Vmax,observed + ωo Ro sin `
(5)
Combining equations (2) and (5) we get
Vgas = Vmax,observed + ωo R
(6)
From other measurements, astronomers have determined that ωo Ro = 220 km s−1 . Finally, rearranging equation (1) we see that the mass of the Milky Way is given by
MR =
2
RVgas
.
G
(7)
Thus, measuring the maximum velocity of the gas and combining this with equations (2) and (6)
will allow us to determine the mass of the Milky Way.
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Figure 2: SRT screen shot.
Procedure
Each group of two students will obtain data at three or four different points in the plane of the
Galaxy. You will share this data with other members of the lab section, so that you will end up
with 10 data points to analyze. To obtain a data point, use following procedure:
1. Set the telescope to scan a range of frequencies in order to obtain the spectrum. To do this,
click on the freq button on the top of the SRT control panel (see Figure 2) and type in the
lower panel either: 1420.4 50 (for the analog receiver; the center frequency and number of
frequency bins); or 1420.4 4 (for the digital receiver; the center frequency and observing
mode).
2. Move the telescope to the Galactic longitude specified by your TA. For example to move
to ` = 10◦ , click on GL10 in the Elevation/azimuth plot in the middle of the screen. The
’Status’ box (middle of screen) will switch to ’slewing’ to indicate the telescope is moving.
When the telescope has completed its move, the icon you clicked on will switch colors and
the ’Status’ box will say ’tracking’
3. Click the clear button on the top left of the SRT control panel to clear the old spectrum and
to start acquiring new data. After several seconds, the spectrum will appear red, in the ’av.
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Figure 3: Sample spectrum obtained at Galactic longitude of 40 degrees.
spectrum integ.’ box in the top middle of the screen (Figure 2). This display will be updated
as the telescope continues to acquire data. Obtain data for about 5 minutes.
4. Left click the mouse button on the average spectrum. This will pop up a new screen, showing
the spectrum. A sample is shown in Figure 3. You can enlarge the window using the mouse.
5. When you are confident that you have a good signal, print the spectrum. To do so, right click
on the camera icon in the system tray (bottom right of the screen), go up the menu to select
’Window/Menu’, and then click on the spectrum window. Do this several times, so that you
can give copies to other people in your lab section.
6. Repeat steps 2 - 5 as needed.
7. For each spectrum, estimate the maximum velocity and error in this measurement. Make a
table of your results (similar to Table 1).
Analysis
Complete your results table by calculating the distance to each cloud, R, ωo R and Vgas . Make a
graph of Vgas as a function of distance from the Galactic center, R. Include the error bars on your
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Table 1: Results Table
Measured
Galactic
Max. velocity estimate error
longitude (deg)
(km/s)
(km/s)
0
10
20
30
40
50
60
70
80
90
Calculated
tangential
ωo R
Vgas
distance (kpc) (km/s) (km/s)
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plot. For each radius for which you have obtained a good measurement, calculate the enclosed
mass of the Galaxy (using equation 7). When you make this calculation, make sure you use an
consistent set of units – it is best to convert everything into meters, kilograms and seconds when
doing the calculation. After you have finished the calculation, convert your calculated mass from
kilograms into solar masses. Plot enclosed mass in solar masses (MR ) as a function of R.
Q1: The analog receiver observed 50 frequency bins, centered at 1420.4 MHz, with a spacing
of 40.0 kHz. What was the starting and stopping frequency of your observations? Using Mathematical Insight 5.1 what was the starting and stopping wavelength of your observations? Using
Mathematical Insight 5.3, what is the range of velocities you could possibly observe?
Note that Mhz ≡ MegaHertz is millions of cycles per second (106 s−1 ), while kHz ≡ kiloHertz is
thousands of cycles per second (103 s−1 ).
Q2: How does your observed mass of the Milky Way compare to the values in the textbook (near
the end of Section 19.1)?
Q3: What do you estimate is the error your mass estimate of the Milky Way? A complete answer
will include a discussion of how you estimated the error in your determination of the maximum
velocity at each Galactic location.
Q4: The average matter density ρ is simply the mass divided by the volume. The volume of a
sphere is given by V = 4πR3 /3 and so
3MR
ρ=
4πR3
Combining the above equation with equation (7) we see that
2
3Vgas
ρ=
4πGR2
(8)
Calculate the average matter density ρ as function of radius R in units of solar masses per cubic
parsec and graph the results.
How does the average density vary with radius?
Q5: What can you conclude from the observed spectrum of the Galactic Center?
Q6: How do you explain the sudden transition from the lack of velocity spread in the H-line at
Galactic longitude less than 20 degrees to a double peaked spectrum at longitudes greater than 20
degrees?
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Radio Observing
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Writing up the Lab
When you complete the lab, have the TA sign your data sheets (spectrum printouts), which will
form part of the lab report. Lab reports are due 1 week after you complete the lab. Lab reports
are to be put into the box (labeled with your TAs name) which are located to the left of the main
stairs when you enter Wilder Lab.
Make sure you follow all of the general guidelines for A3 lab reports, which are discussed in a
separate document posted onto Blackboard. You should use complete sentences throughout your
lab report. When answering questions, please do so in an essay style, referencing the question you
are answering. When you have equations to solve, show all of your work (not just the answer), and
include units if needed. Make sure you show all details of your calculations, and describe how you
obtained your results.
Pre-Lab Questions
1. How many meters are there in 8 kpc?
2. If I determine the line of sight velocity of a cold neutral hydrogen gas cloud to be 150 km/s,
at what wavelength do I observe the hydrogen emission line?
3. If I point the radio telescope at a Galactic Longitude of ` = 40◦ , at what distance from the
Galactic center will I measure a maximum velocity for the gas? Express your answer in
parsecs.
4. The Student Radio telescope has a diameter of 2.1m. What is the angular resolution of this
telescope when observing the 21cm line of hydrogen? How does this compare to the angular
resolution of the human eye?