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# Measurement of the Milky Way Rotation Abstract : In this post we want to continue the study of the structure of our galaxy made using the emission at 21 cm of neutral hydrogen and described in the previous post : Milky Way Structure detected with the 21 cm Neutral-Hydrogen Emission. In particular, we want to exploit the Doppler effect to measure the rotation speed of neutral hydrogen clouds, as a function of the distance from the center, and derive the rotation curve of the galaxy.

### Introduction

Neutral hydrogen gas in the disk of our Galaxy moves in nearly circular orbits around the Galactic center. Radial velocities Vr measured from the Doppler shifts of HI λ=21 cm emission lines encode information about the distances of Hi clouds from the galactic core. HI is transparent to microwave radiation except in a few regions near the Galactic plane, so the distribution of hydrogen maps out the large-scale structure of the whole Galaxy, most of which is hidden by dust at visible wavelengths.
The image below shows the diagram of our galaxy with the position of the sun. The sun is located in a secondary arm, called the Orion arm, between the arms of Perseus and Sagittarius. Rotation occurs clockwise. ### Doppler Shift

The Doppler effect (or the Doppler shift) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. We are familiar with the changing pitch of a moving vehicle. For light, this change is given by : where c is the speed of light, v the relative velocity, λe the emitted wavelength, and λo the observed wavelength. Using the relationship λ=cf, we can write the relation in terms of frequency and calculate the relative velocity : We know the emitted frequency because it is measured in laboratory fe=1420.40575 MHz, we know the speed of light and we measure with the radio telescope the observed frequency, so we can calculate the relative speed between the observer (the radio telescope) and the hydrogen cloud.

### Tangent Point Method The measurements are made by pointing the radiotelescope towards the first quadrant of the galaxy, with increasing galactic longitude, starting from values ​​close to 0° (direction towards the galactic center) to arrive at values ​​close to 90° (direction parallel to the direction of rotation).
We refer to the diagram on the side where the position of the radio telescope (coinciding with the sun) and the direction of observation are shown, the vector of the speed of the sun is highlighted in yellow and the vector of the speed of the hydrogen cloud in blue. As can be seen from the drawing, the highest recession velocity measured in that specific direction is the one that corresponds to the clouds that orbit tangent to the direction of observation, of course there may also be other contributions from closer or more distant clouds but, with respect to the point of observation (the sun) they will have ever lower recession.
By measuring the frequency spectrum and calculating the relative speed with the Doppler effect formula, we obtain the velocity distribution of the hydrogen clouds and, among these, we must take the highest value corresponding to one of the relative maxima. This measurement procedure must be repeated, as we said above, for increasing galactic longitudes, in order to obtain a mapping of the entire first quadrant.
The value we are interested in, i.e. the tangent speed of the cloud with respect to the galactic center Vtp/gc, must however be calculated by adding the value obtained from the measurement, which is the value of the tangent speed of the cloud with respect to the sun, Vtp/s, with the value of the rotation speed of the sun along the direction of observation, Vs/gc.

With a bit of trigonometry we can write :

Vtp/gc = Vtp/s + Vs/gc = Vobs + Vsun*cos(90 – Θ)  = Vobs + Vsun*sin(Θ)

d = Ro*sin(Θ)

Where :
Vsun = 220 Km/s
Ro = 7.6 Kpc
Θ = galactic longitude angle
Vobs = speed calculated from measurements with the radio telescope

### Correction for Local Movements

Of course things are not so simple … The speed value we obtain by applying the Doppler effect formula, must be corrected by removing the contribution of three movements that affect the movement of the observer (our radio telescope), compared to the “simple” rotation around the galactic center :
– the rotational motion of the earth
– the revolution motion of the earth around the sun
– the motion of the sun within the local group of stars

When we took into account these three movements we placed ourselves in what is called: local reference system (local standard rest). To obtain the value to remove (or add) you can refer to the following link : https://www.gb.nrao.edu/cgi-bin/radvelcalc.py

### Measures

Those shown below are the graphs of the speeds we obtained with our radio telescope. The values ​​always refer to the “standard local rest”. For each measurement we report galactic longitude and latitude. Latitude is always close to 0 °. The velocity distribution was approximated by four Gaussian curves, for which we report the average velocity value corresponding to the maximum of the curve. In our simplified hypothesis, the Gaussian curves should correspond to as many hydrogen clouds moving with respect to the observer.

The first curve that we report differs from the following ones because it corresponds to the observation made at a longitude of about 180 °, that is towards the outside of the galaxy in the opposite direction to that towards the center. If our rotational hypotheses are correct, the measured relative speed should be practically zero since the radial component of speed is zero. The measurement agrees excellently with the predictions since we got just a zero speed value, as can be seen in the graph below. Subsequent measurements refer to increasing values ​​of galactic longitude, starting from an initial value of 12°. As can be seen from the graphs, the main peak, which should correspond to the local clouds present in the Orion arm, remains at speeds always close to zero, this is consistent with the fact that they are local clouds rotating at the same speed as the sun. As longitude increases, the contribution of components with greater redshift increases, until it reaches a maximum at about 45 °, and then decreases as longitude increases further. These higher redshifts should correspond to emissions from clouds in innermost locations that are away from the sun.         ### Galaxy Rotation Curve

With the data collected in the observations we have prepared the table below in which we insert the galactic longitude, the measured speed value, the value of the sun’s speed along the measurement direction, the speed Vtp resulting from the sum of the two components and the distance corresponding to longitude, calculated with the trigonometric formula. The data in the table are shown in the following graph which represents the trend of the rotation speed of the galaxy as a function of the distance from the center. Our measurements have a great uncertainty due above all to the imprecision of the radiotelescope pointing system and to the poor angular resolution, moreover the frequency resolution, although good, does not always allow a clear separation of the different components that make up the spectrum.
Despite these limitations, the result is appreciable because it clearly shows an initial increasing section, followed by a plateau with values ​​that average around 225 Km/s. This trend is very similar to the rotation curves found in the scientific literature, we report an example in the next graph.  The rotation models of the galaxy, based on the estimate of the mass distribution, predict that the rotation speed decreases as the distance from the center increases. The estimate is based on observations showing that most of the galaxy’s mass is concentrated in the interior. In this configuration a “Keplerian” model is applied (a bit like the solar system which has its mass concentrated in the sun) which predicts that the rotation speed is proportional to 1/√r, that is, it decreases as the distance increases.
The data, on the other hand, say that the speed remains more or less constant even as the distance from the center increases.
This anomalous behavior, common to most galaxies, is an indication of the existence of the mysterious dark matter, which would be distributed in a large halo around the galaxy. Dark matter would be the missing mass needed to explain the “anomalous” rotation curves of most galaxies.

### Conclusions

The self-built 21cm radio telescope has allowed us to obtain important scientific evidence on the structure and rotation of the galaxy. The limited angular resolution is not a big problem because the hydrogen clouds are distributed objects, while the good frequency resolution has allowed us to study in detail the relative movements and rotation of the hydrogen clouds. All this was possible thanks to the availability of cheap LNAs and filters and the use of the SDR technique for data acquisition and post-processing. The GNURadio open software also made it possible to use and write DSP data processing programs.

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