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: ... [but I can't remember where I took this plot from :-( ])

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 JAVA applet below 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.

Some more explanations for the Applet

How to determine the rotation curve of our Galaxy

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.

Observational Procedure

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:

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!

Batch Observations:

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

Analysis, Step One: Basic data reduction

Let's suppose that you use Microsoft Excel to do the interpretation of the data. Then we recommend to follow this sequence (see also here):

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 ...

Analysis Cosmetics: Optional improvement of the data representation

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)!

Analysis, Step Two: Get the rotation velocites

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:

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

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

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

Gal.Long.v_sun*sin lVmaxv(R)R=R_sun*sin l

Analysis, Step Three: Interpretation

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!

Analysis, Step Four: Modeling

If you want to compute the rotation curve from a mass model of the Galaxy and explore yourself the possible explanations, here you can download instructions for modeling in Excel.

Analysis: Another way of looking at the l-v-map

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

V_rad = v_rot - v_sun * 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 v_rot = 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.

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last update: Nov. 2010 J.Köppen