Deriving the Galactic Rotation Curve

Joachim Köppen Strasbourg 2010

The gas and the stars that make up the disk of our Milky Way Galaxy revolve around the Galactic Centre. This finding comes from optical studies of stellar motions and from observations of the 21 cm radio line which permits to measure the motion of atomic hydrogen gas by the Doppler effect. Below we show the rotation curve of the Milky Way, i.e. how the rotational speed depends on the distance from the Galactic Centre. (modified after: Fich & Tremaine 1991, Ann. Rev. Astron. Astrophys. 29, 409)

Because we observe the motions from within the Galactic Disk - the Sun is located in the Plane at about 8.5 kpc distance from the Centre - and the Doppler effect measures only the radial component of the velocity, the structure of the observed data is a bit complicated. The JavaScript Radial Velocity Tool visualizes how we perceive the true galactic motions. Here are basic explanations:

The image below shows a map of the radial velocity which every part of the Galactic Disk appears to have as seen from the Sun. Since the Sun also participates in the rotation, we observe only differences in radial speed. Regions coloured in red appear to be receeding from us, hence the line is redshifted (i.e. to frequencies below the theoretical frequency); blue regions appear to move towards us (blueshift). The intensity of the colour is a measure of the apparent radial velocity. Emission from the greenish-greyish regions is seen without any lineshift, it appears to move with the same speed as ourselves.

If we look in a certain direction in the Plane - the black line in the above image indicates this line of sight at galactic longitude of 50°, with small dots indicating distances from the Sun (every 5 kpc) - the radial velocity of a parcel of gas depends on the distance from the Sun to that gas:

As we can already see from the map, at that direction all the gas up to distances of about 10 kpc has positive radial velocity (redshift); the matter further away is blueshifted.

One important feature is seen here: for the gas inside the solar orbit the radial velocity has a maximum value. The dependence of this maximum on galactic longitude permits to derive the Galactic Rotation Curve (cf. below). But the fact that for each value of the radial velocity there are two distances where the emitting gas could be, makes it a bit difficult to locate this gas. This problem does not exist for the regions outside the solar orbit.

With the ESA-Haystack radio telescope we can derive the rotation curve of the Galactic Disk by obtaining spectra at positions in the Galactic Plane for galactic longitudes up to 90°, measuring the radial velocities of the hydrogen gas, and using the trick of the maximum velocity to infer the rotational speed at various distances.

Since only galactic longitudes up to 90° are of use for us, it is best to check whether this part of the sky is above the horizon ... BEFORE going to the observatory room! This graph shows when positions in the Galactic Plane are accessible:

Evidently, observations are possible only between about 14:00 and 02:00 local sideral time (LST). What this means in terms of civil time, depends on the time in the year: at the end of july, the local noon (12:00 CEST) in Illkirch will be at 08:00 LST - this means that stars with 8hrs right ascension will pass through the southern meridian (this for instance the constellation of Cancer). Therefore, you can start observations about 6 hours later, and spend the evening observing!

Once you started up the system, as described here you are ready to observe. There are two ways to get the observations done. We recommend to do them first manually. This direct interaction will make you more familiar with the instrument and will teach you how to use it well. On the graphical interface, watch the waterfall plot. The variation you see between each line should be slight and is due to the fluctuations of the noise in the signal. You may click on the yellow fields to adjust the range of the values represented by the colours. Since the galactic emission is concentrated to a narrow frequency range, you should be able to discern eventually a vertical band of slightly higher signal. This is also seen in the frequency plot to the upper centre: the black curve shows the current spectrum, the red curve is the accumulated one, so that after a while the galactic features would become more distinct. After you had tried the manual observing, we will encourage you to run batch files while you are in class or doing other things. This will allow for much easier, less tedious data acquisition and hopefully permits you to accumulate as much data as you may need.

Manual Observations:

• Since only galactic longitudes up to 90° are of interest, find those points which are currently visible and not close to the horizon. Usually, positions in the Galactic Plane with longitudes l=10 ... 90 should be present in the catalog and they are marked as G10 ... G90
• Make sure that your frequency is centered at 1420.4 MHz and your width is 1500 kHz span. Also, if not already done, plug in the PointCorr: Az =-0.5 El = -3.5 which are corrections necessary for the proper positioning in the sky.
• To avoid forgetting to switch on the recording of data, click on the Record button at the start of your run. All the data will be recorded and later you can easily sort them out!
• Move the telescope one of the positions available by clicking on that point. The name of the source will be displayed on the right hand panel, and the telescope will execute the move. The actual position is displayed on the top right, with a background that blinks red (while moving in azimuth) and blue (elevation).
• When it is finished moving, take some time until you see the galactic feature, somewhere in the middle of the spectral range. Since we shall be primarily interested to determine maximum radial velocity, make sure that the low frequency end of the feature is well in the spectral range. If not, you can change the central frequency by entering it in the corresponding field and hitting the return key. At the lower right, an appropriate message should appear in blue letters.
• BTW: Everytime you give a command, record it in the logbook, as well as any comments that you may have on the spectrum.
• Stay on that position, until the galactic feature shows up nicely as the red curve of the averaged spectrum at upper centre. When this is the case, you can also click on the leftmost structure of the feature and read off (at lower right, in blue letters) its radial velocity
• You can also double-click on this frequency plot: a new window will open and show the averaged, baseline-subtracted spectrum, converted into radial velocities. What is the highest velocity where you can still see some emission from the gas?
• Go to another position, and repeat these operations for all positions available
• After your last observation, stop recording by pressing the Record button again.
• in case neither you nor someone else wants to use the telescope, stow it by pressing Stow on the top bar. Then shut down the system as described here.
• You find your recorded data in the text file at: C:/SRTcassi, named: year/month/day/time.txt for example: 1004141630.txt if I observed starting at 16:30 on April 14th 2010 (all times are given in UTC). Copy the file to your USB stick or flash drive to transfer onto your computer.

You may wish to observe also positions with longitudes in between those marked (usually in 10° intervals). You can do this, but it needs adding or modifying sources in the software's catalog file. This is not terribly complicated and it can even be done during an observation run ... but please think of other users of the telescope, and after your run, please remove or undo your changes!

• in the folder C:/SRTcassi/ you find the file srt.cat
• make a security copy of it, just in case ...
• open the file with a simple editor, Blocknotes or WordPad (click the file with the RIGHT mouse key)
• the syntax of this file is described in the software's Help window. but if you want to add G12, simply put in a line
GALACTIC 12 0 G12
Hint: it is best to add any sources BEHIND all the other sources ... However, you will find that this new source is too close to the existing point of G10 ... in this case it is better to modify the existing entry for G10 to G12 ... store the changes to the file
• when you are doing this while the software is running, you click the reReadCat button, and you'll see that your new position is displayed. Note: if you are also observing a source, you might find that the telescope will go to a different source: this rereading has altered the numbering of the sources!

Batch Observations:

Unfortunately, during summer, the galactic positions interesting for the rotation curve are only above the horizon at night. Therefore,

• we have prepared batch files to do all the positions G0 to G90 in one go, and the observations will only start when the Galactic Centre appears above the horizon at about 15:00 LST (which is about 20:00 CEST in late July). Until then, the telescope will do nothing ... so there is no point to block the telescope for other users!
• These batch files are found in the folder SSP10, and they are
• GalRotationDown_LST15.txt which takes about two hours
• GalRotationUp_LST16.txt which also takes two hours
• NightSurvey_LST17.txt which runs all night for more than eight hours
• Anytime before the scheduled start, click cmdfil and open the appropriate .txt file ... the rest will be done by the software and the telescope
• In your first batch observations you might want to follow what is going on for a while. As each position takes about some minutes, you have time to observe how well the features in the red curve build up. The final spectrum will be close to what you will obtain in your analysis.
• After the batch is completed, you can find your recorded data in the textfile at: C:/SRTcassi, named: year/month/day/time.txt for example: 1004141630.txt if I observed starting at 16:30 on April 14th 2010 (all times are given in UTC). Copy the file to your USB stick or flash drive to transfer onto your computer.

• To import your text file go to: File -> Import-> Text File -> Select your Text File -> Delimited Text file -> Delimiter is spaces -> Finish
• Each row represents a single spectrum taken by the telescope at a certain time instant. The columns from left to right represent: Time, Azimuth, Elevation, offsets in azimuth and elevation, velocity with respect to the Local Standard of Rest (LSR), the first frequency in the spectrum, the frequency increment, an integer number indicating the frequency mode, the number of frequency points in the spectrum, followed by the flux at the first frequency all though to the flux at the last frequency.
• Furthermore, there will be rows with messages from the software. Some of them indicate the source at which the following spectra were taken:
* G20 .................. from manual observations
* cmdfil: line 52 galactic 20 0 ................... from batch observations
• After each of these messages, insert at least six new rows, to separate the data chunks for each galactic longitude. At each of these data chunks do the following:
• in the first of the new rows, make an array of frequencies: in the column K which contains the flux of the first frequency put in the start frequency (from column G), then fill the rest of the array by adding the fequency step (from column H) to the preceding frequency.
• in the second row, make an array of radial velocities: use the formula
vrad = -(f/1420.406 - 1)*299796 - VLSR
where VLSR is taken from column F of the data. The value of VLSR is the same for all spectra taken at that particular position: you may notice a small variation, since our position rotates with the Earth; but this is negligible -- it is completely sufficient if you simply take the value from the first row, that is the first spectrum taken at this position!!
• in the third row, make an array of average fluxes, by putting the AVERAGE of the flux values from all the data chunk's cells in each column.
• now make a plot of the average fluxes as a function of frequency (or radial velocity, if you like). This would like this
• from this plot you can already read the highest radial velocity (or the lowest frequency) of the galactic feature ...
For example, the averaged spectrum at G30 looks like this (on 24 mar 2010, with 30 min observing time, resulting in about 140 spectra ... this shows that our system isn't quite optimal yet). While using the frequency as abscissa would also do, it is much better to use radial velocity:
You see that the galactic emission extends up to a maximum radial velocity of about +120 km/s (indicated by the vertical black line). At this point, you are ready to extract this essential information about the rotation curve, as described in Step Two. But if you like to polish up a bit your analysis, you can do so as described in the following ...

Often it is advantageous to improve the appearance of the galactic features by subtracting the baselines from the averaged spectra of each galactic longitude. As described here it requires inspection of each spectrum as to find the best way to fit a baseline to the background, and then to subtract these constant or interpolated baseline fluxes from the spectra to give the spectra of only the galactic feature:

With these spectra you can determine the maximum radial velocities even more reliably. The spectrum below is the the same one shown above, taken at G30, but now baseline-subtracted and a smoothing over the 4 frequencies was applied:
As you can see, the curve looks nicer now, but the maximum radial velocity is still about +120 km/s (marked by the black vertical line)!

The next step is to determine for each of your observed positions the maximum radial velocity, by looking at the averaged spectra. These values permit you to derive the rotational speed in the Milky Way at various distances from the Centre.

This is how it is achieved: Let us assume - as in the JAVA applet - that all parts in the Galactic Disk are moving in circular motion around the Galactic Centre, and that the rotational speed depends in some way on the distance from the centre. If we look from the Sun into the direction of galactic longitude l we observe on our line of sight a parcel of gas with radius R from the centre, we measure its radial velocity with respect as its circular velocity v(R) projected on the line of sight, with the projected velocity of the Sun subtracted:

Vrad = v(R) * sinδ - vsun * sin l
as illustrated below:
From the rules of the sines in a triangle we get R sinδ = Rsun * sin l which gives
Vrad = (v(R) * Rsun/R - vsun) * sin l
The maximum radial velocity is measured when the line of sight is tangential to the circular orbit: R = Rsun * sin l which gives
Vmax = (v(R)* Rsun/R - vsun) * sin l = v(R) - vsun * sin l
and the simple relation
v(R) = V_max + v_sun * sin l

This is the crucial formula which transfoms our measured values Vmax (l) determined for each longitude l into the rotational speed v(R) at the galactocentric distance R = Rsun * sin l .

Let's do it: We may take the standard accepted values for the Sun: Rsun = 8.5 kpc and vsun = 210 km/s and put our measured maximum radial velocities in a small table (in Excel) - with the data from 24 mar 2010 -

 Gal.Long. vsun*sin l Vmax v(R) R=Rsun*sin l 10 36.5 150 186 1.48 20 71.8 138 209 2.91 30 105 120 225 4.25 40 135 94 229 5.46 50 160.9 73 234 6.51 60 181.9 52 234 7.64 70 297.3 33 230 7.99 80 206.8 23 230 8.37 90 210 17 227 8.5

We now plot the rotation curve - rotational velocity as a function of distance R from the centre. This simple plot shows one of the great problems of our present knowledge:

In order to explain the constancy of the rotational velocity the amount of visible matter (stars, gas, and dust) in a galaxy does not suffice. So far, the best idea is to postulate the existence of an invisible Dark Matter whose presence is only evident by its gravitational attraction. However, to account for the rotation curve of the Milky Way (and the same holds for other spiral galaxies as well) one needs about ten times more dark matter than there is visible matter!

We may also obtain a direct interpretation of the longitude-radial velocity map by superposing the curves of constant rotational speed: Using the formula that gave us the maximum radial speed

Vrad = vrot - vsun * sin l
we plot this speed as a function of longitude for a constant value of v_rot. In the map below the curves refer to vrot = 250, ..., 200 km/s (from top to bottom).
Depending on the signal level that we decide to mark the maximum velocity - for instance the border between blue and violet - we see that for longtitudes greater than 30° the rotational speed is around 240 km/s. In the innermost part the speed drops to 220 km/s (at 30°) and 200 km/s (at 10°), where the rather weak emission makes it a bit difficult to define reliably what the maximum radial speed could be ... obviously more sensitive observations are needed.

last update: Nov. 2010 J.Köppen