Ph 343 --- Lab 6
The Galactic Rotation Curve

A. Purpose: Measure the rotation curve of our Galaxy interior to the Sun's distance from the Galactic center. Also, place limits on the circular velocity at the Sun's distance from the Galactic center. The measured rotation curve will yield the amount and distribution of mass in our Galaxy from the center out to the Sun's location.

The data for the lab is observations taken last fall at a range of galactic longitudes, $\ell$, and new observations taken at $\ell = 90$ degrees.

B. Background: The methods for determining the kinematics and mass of the Milky Way Galaxy are described in section 22.3 of Carroll & Ostlie. We will determine the Galactic rotation curve using the standard method of measuring the radial velocities of HI gas along lines of sight with galactic longitudes $0^\circ < \ell < 90^\circ$ ($1^{st}$ quadrant - which is best observed from the northern hemisphere) and $270^\circ < \ell < 360^\circ$ ($4^{th}$ quadrant - best observed from the south). In these quadrants, the most positive (1$^{st}$) or negative ($4^{th}$) radial velocity occurs for gas at the point where the line of sight penetrates closest to the galactic center. This point is called the tangent point and occurs at a distance from the Galactic center of

\begin{displaymath} R_{min} = R_0 \sin(\ell), \end{displaymath} (1)

where $R_0$ is the Sun's distance from the center. The radial velocity of the gas at the tangent point is
\begin{displaymath} v_{r,max} = \Theta(R_{min}) - \Theta_0 \sin(\ell), \end{displaymath} (2)

which implies
\begin{displaymath} \Theta(R_{min}) = v_{r,max} + \Theta_0 \sin(\ell). \end{displaymath} (3)

Here $\Theta_0$ is the circular velocity at $R_0$. Observations at a series of galactic longitudes in the range $0^\circ < \ell < 90^\circ$ will map out the circular velocity for $0 < R < R_0$.

One problem with Equation [*] is the presence of $\Theta_0$, which the observations do not yield. One approach to determining $\Theta_0$ is to measure the spectrum of HI emission at $\ell = 90^\circ$. At that longitude,

\begin{displaymath} v_{r} = \Theta(R)(R_0/R) - \Theta_0. \end{displaymath} (4)

If the disk of our Galaxy extends to sufficiently large radii and $\Theta(R)$ does not increase faster than $R$, then the most negative observed radial velocity will be $-\Theta_0$.

Because the HI gas in the galactic disk is on nearly circular orbits, the mass of the Galaxy interior to any radius $R$ is given by the usual formula for circular motion:

\begin{displaymath} M(R) = \frac{\Theta(R)^2 R}{G}. \end{displaymath} (5)

Here $G$ is the gravitational constant, equal to $6.67 \times 10^{-8} {\rm cm}^3 {\rm g}^{-1} {\rm s}^{-2}$. In units that are more useful for Galactic dynamics, $G = 4.30 \times 10^{-6} {\rm kpc} ({\rm km/s})^2 (M_\odot)^{-1}$. $M_\odot$ is the solar mass ( $1.989 \times 10^{33}$ g). The above formula assumes that the mass of the Galaxy is spherically distributed. This is not very accurate for the region between the Galactic center and the Sun - the flat disk contains at least a significant fraction and perhaps most of this mass - but the correction for the flattened distribution of the mass is on the same order as the uncertainties in the rotation curve.

C. Observations:
Log onto GreenBank and start the SRT control window. Calibrate with the noise diode at an altitude of at least 30 degrees. Use a frequency of 1419.4 MHz, 50 frequency bins, and a zero frequency step.

For the observations of Galactic HI, use a central frequency of 1420.6 MHz and 60 frequency bins with the default spacing of 0.04 MHz. This places the highest frequency observed at 1421.76 MHz, which is the frequency that HI with a radial velocity of -287 km s$^{-1}$ would have. Since $\Theta_0$ is about 220 km s$^{-1}$, this should give us an adequate number of points with which to determine the baseline on the high-frequency end of the scan.

Point the telescope to $\ell = 90^\circ$, $b = 0^\circ$ and record 6000 s of data. Repeat at $\ell = 90^\circ$, $b = +15^\circ$. This second observation is to check for the effects of the warp of the Galactic disk.

D. Data:
The maximum radial velocities obtained by the class in the direction of the different longitudes are given in the table below. The first column is the galactic longitude, the second and third columns are the lowest observed frequency with HI emission and its estimated uncertainty, the fourth and fifth columns are the corresponding observed maximum radial velocity and its uncertainty, and the final column is the number which should be added to the observed radial velocity to correct for the motion of the Earth around the Sun.


Table: Measured Maximum Radial Velocities
longitude $\nu_{min}$ unc $\nu_{min}$ $v_{r,max}$ unc $v_{r,max}$ $v_{helio}$
(deg) (MHz) (MHz) (km/s) (km/s) (km/s)
5 1420.08 0.04 68.9 8.5 -15.4
10 1419.96 0.04 94.2 8.5 -16.5
15 1419.64 0.04 161.8 8.5 -17.4
20 1419.72 0.04 144.9 8.5 -17.5
25 1419.76 0.04 136.4 8.5 -18.3
30 1419.76 0.02 136.4 4.2 -23.6
35 1419.80 0.02 128.0 4.2 -19.1
40 1419.84 0.04 119.5 8.5 -23.2
45 1419.92 0.02 102.7 4.2 -20.1
50 1419.92 0.02 102.7 4.2 -22.4
55 1419.96 0.04 94.2 8.5 -21.7
60 1420.08 0.02 68.9 4.2 -21.0
65 1420.16 0.02 52.0 4.2 -19.0
70 1420.16 0.04 52.0 8.5 -17.4
75 1420.20 0.04 43.5 8.5 -17.1
80 1420.16 0.02 52.0 4.2 -16.7
85 1420.20 0.04 43.5 8.5 -16.1

The observed radial velocities also need to be corrected for the motion of the Sun with respect to the local standard of rest (LSR). Recent measurements (Dehnen, W. & Binney, J.J. 1998, Monthly Notices of the Royal Astronomical Observatory, Vol. 298, p. 387) suggest that the components of the motion of the Sun with respect to the LSR (the Sun's ``peculiar velocity'') are $10.00\pm 0.36$ km/s towards the Galactic center ( $\ell = 0^\circ$), $5.25\pm 0.62$ km/s in the direction of the Sun's rotation around the Galactic center ( $\ell = 90^\circ$), and $7.17\pm 0.38$ km/s towards the north Galactic pole ($b=90^\circ$). To correct a radial velocity to that which would be obtained by an observer moving with the LSR, calculate the component of the Sun's peculiar velocity along the line of sight and add it to the radial velocity corrected for the Earth's motion around the Sun.

E. Analysis: estimate of $\Theta_0$
The goals for the first part of the analysis are to produce HI spectra with a sloping baseline subtracted and to measure the most negative velocity with HI emission in your spectra from $\ell = 90$ deg.

1. For each of your two pointings, calculate the average system temperature at each observed frequency. Also calculate the uncertainty in that temperature.

2. Use the same procedure as in Lab 5 to fit and remove a linear baseline.

Your report should include a printout of the lines from the spreadsheet that show your calculations, but it does not have to have a printout of the raw data.

3. Plot antenna temperature vs. frequency for the results of part 2. Include error bars. Assume that the uncertainty in each antenna temperature is the uncertainty in the original average system temperature. This neglects the uncertainty in the fitted baseline. Discuss how good a job the linear baselines did at producing a subtracted spectrum with temperatures constant at 0 for frequencies outside of the range with HI emission.

4. For each of your two plots, find the highest frequency bin with an antenna temperature significantly greater than zero. Is there a significant difference in the highest frequencies with emission for the $b = 0^\circ$ and $b=15^\circ$ spectra? What could cause a difference?

5. Use the Doppler formula, $v = c(\nu_0 - \nu)/\nu_0$, to calculate the radial velocity that corresponds to each of the highest frequencies. The laboratory frequency for the HI emission is $\nu_0 = 1420.4$ MHz. Use your judgment to estimate the uncertainties in these radial velocities and justify your values.

6. Correct the observed velocities found in part 5 for the motion of the Earth and the Sun and thus obtain a lower limit on $\Theta_0$. You will need to consult with me on how to find the correction for the motion of the Earth. How do the magnitudes of your corrected velocities compare to the International Astronomical Union standard value of $\Theta_0 = 220$ km/s. Can you give reasons for any difference from the standard value?

F. Analysis: the rotation curve
The second part of the analysis finds the Galactic rotation curve from the observations between $\ell = 0$ deg and $\ell = 90$ deg.

1. Calculate the radius of the tangent point, $R_{min}$, for each observed galactic longitude. Assume $R_0 = 8.5$ kpc.

2. Calculate the correction to the LSR for each longitude and then correct the maximum radial velocity of the HI emission for the motion of the Earth and the Sun. This corrected radial velocity is the one that should be used in the expression for $\Theta(R_{min})$.

3. Calculate the circular velocity for each longitude, $\Theta(R_{min})$. Assume $\Theta_0 = 220$ km/s. Use propagation of errors to derive a formula for the uncertainty in $\Theta(R_{min})$ in terms of the uncertainty in $v_{r,max}$. Use this to estimate the uncertainty for every value of $\Theta(R_{min})$. Plot your rotation curve ( $\Theta(R_{min})$ vs $R_{min}$), including your uncertainties, as well as listing your calculated values.

4. Use your calculated values for $\Theta(R_{min})$ to calculate the mass of the Galaxy interior to each $R_{min}$, $M(R_{min})$, in solar masses.

5. The radial velocities $v_{r,max}$ in Table [*] are the greatest radial velocity with significant HI emission. This velocity probably slightly overestimates the radial velocity of the HI gas at the tangent point because turbulent velocities in the gas produce some emission at velocities larger than the average, bulk motion of the gas. Is there any evidence for such turbulent velocities in your rotation curve? (Hint: Consider the values of $\Theta(R)$ for radii near $R_0$.)