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Determination of Milky Way Rotation Curve Through Observation of Redshift of 1420.40 MHz Radiation

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Determination of Milky Way Rotation Curve Through Observation of Redshift of 1420.40 MHz Radiation

  1. 1. Determination of Milky Way Rotation Curve Through Observation of Redshift of 1420.40 MHz Radiation Daniel A. Bulhosa∗ MIT Department of Physics (Dated: September 8, 2014) In this experiment we used a 7.5 foot diameter parabolic dish and receiver designed by the Haystack Observatory to determine the velocity curve of matter in the disk of the Milky Way. This was accomplished by measuring spectra centered at the atomic hydrogen hyperfine transition line of 1420.40 MHz and determining the velocity corresponding to the highest observed redshift. Our measured values show reasonable agreement in the moderate to large radius regime with recently published values. Consideration of the amount of matter contained within 8.33 kpc of the galaxy, as suggested by the rotation curve, reveals a discrepancy between the observed amount of matter and the matter necessary to create the measured orbital speeds of matter around the galaxy. I. INTRODUCTION Until the mid 20th Century our knowledge about the Milky Way came from observations of the visible light generated by the stars it contains. For example, lit- tle was know about the dust and gas between the stars whose motion we measure in this experiment (known as the Interstellar Medium, or ISM), even after its existence was proven by R.J. Trumpler in 1930 [1]. Then in 1932 Bell Laboratories employee and radio engineer Karl Jan- sky discovered a periodic noise while carrying out exper- iments related to the maximization of the signal to noise ratio in radio communication [2]. Although Jansky be- lieved at first that the spike might be due to the transit of the sun across the celestial sphere, further analysis led him to conclude that the peak of the noise was measured when his device was pointed along the galactic plane in the direction of Sagittarius. Jansky’s discovery marked the birth of radio astronomy. Physicist Jan Oort was one of the first people to rec- ognize the importance radio emissions in the study of the structure of the Milky Way. He prompted another younger physicist, Hendrik van de Hulst to look for an emission line that would be useful for radio astronomy. Hulst proposed the detection of ground-state atomic hy- drogen through its emission of 21-cm (1420.40 MHz) ra- diowaves when its electron undergoes a hyperfine tran- sition. Detection of ground-state atomic hydrogen is an attractive pursuit as it is the predominant element in the galaxy. Furthermore—partly because of its long wave- length relative to the size of the atomic components of the ISM—the associated 21-cm radiation undergoes lit- tle extinction when travelling across interstellar space, so that intervening matter does not obscure distant objects within the galaxy. In this experiment we used a small radio telescope (SRT) to determine the rotation curve of the Milky Way and learn about its matter distribution. ∗ II. THEORY The Milky Way is composed of a spheroidal component and a flat disk component. The spheroidal component has a radius of 30kpc or more, and has a bulge at the center of diameter of about 3 kpc. The disk has a radius of about 30 kpc and is composed of gas and a variety of different tyoes of stars. The orbit of matter in the galactic disk is very nearly circular [3]. Suppose that we point our antenna in the direction of galactic longitude l along the galactic plane (see Figure 1). The beam of our antenna will intersect the circular orbit of the matter in the disk of our galaxy at radius R. Let P be the point of intersection nearest to the Sun, then after application of the sine law one finds that the speed of matter at P relative to the Earth in the direction of the antenna beam is: ∆vb = v(R) R R − vLSR sin(l)−(V +VEarth) (1) Here v(R) is the speed of the matter at radius R, vLSR is the mean speed of the matter in the neighborhood of the Sun, and the last term accounts for the relative mo- tions of the LSR, the Sun, and the Earth. If we make the reasonable assumption that v(R) increases less than lin- early with increasing R then we can see from (1) that for a fixed l the maximum ∆vb occurs when R is minimized— that is, when R is the radius of the orbit to which the antenna beam is tangential. For this R we have by the sine law that R sin(l) = R, so substituting into (1) an rearranging terms we find an expression for v(R) which holds whenever R < R . When we detect electromagnetic emissions from the galaxy within some frequency interval centered at 1420.40 MHz we observe a distribution of frequencies, rather than a single peak at the center. This is due to the motion of the matter in the galaxy, which causes the light emitted by this matter to be Doppler shifted. Be- tween galactic longitudes 0o and 90o the fastest moving
  2. 2. 2 FIG. 1. This figure was adapted from [4]. Here γ = l is the galactic longitude, which measures the angle between the direction of our antenna beam and the center of the galaxy along the galatic plane. objects will be the ones emitting the most red-shifted fre- quencies. The relation between the measured frequency and the relative speed between the source and the Earth along the antenna beam is given by: ∆vb = c 1420.40 MHz − fmeas 1420.40 MHz (2) Combining equations (1) and (2) we can use the maxi- mum redshift measured by our telescope at a given lon- gitude to determine v(R). III. EXPERIMENTAL SETUP A diagram of the apparatus is shown in Figure 2: FIG. 2. A diagram of the different components of the appa- ratus. The left box with the concave down curve drawn in it stands for the first band-pass filter and the LNA. The other similarly drawn box towards the right is the low-pass filter. ADC stands for Analog to Digital Converter and L.O. stands for Local Oscillator. The dish of the antenna is made up of a C/Ku band mesh. Radiation of wavelength at least ten times larger than the size of the surface holes of the mesh is focused by the dish onto the antenna feed [3]. The electromagnetic waves reaching the antenna feed induce an AC signal in it. That signal is pre-ampliflied and then passed through a band-pass filter that transmits a 100 MHz band centered at 1420.40 MHz. The filtered signal is then amplified by the Low Noise Amplifier (LNA), which gives a 25 dB amplification to signals with frequencies between 1400 MHz and 1440 MHz. At the mixer, the amplified signal is combined with an artificial signal from a local oscillator whose frequency f is adjustable. The mixer outputs two signals, one is the original signal that went in with all of the frequencies that compose it shifted down by f, and the other is the original signal with all the composing frequencies shifted up by f. A low-pass filter then removes the second signal, and the remaining signal is digitized. Once digitized the sig- nal, which describes the power per unit time collected by the antenna in the relevant bandwidth, is Fourier trans- formed into a signal describing the amount of power con- tributed by each frequency of radiation. This signal is sent to the SRT software, which organizes, records, and displays the data. A plot of the Fourier transformed raw data is shown in Figure 3. FIG. 3. Plot of the contribution of each frequency bin in the bandwidth of interest to the total power measured. For each measurement set there were 148-156 frequency bins spaced by 7.81 kHz. We can see in this plot the relative attenuation of the frequency signals outside of the 40 MHz bandwidth the amplifier amplifies. The information of interest is the fine structure at the top of the curve shown. We observed a spike in power at 1420.0 MHz. It was determined that the spike was not of natural ori- gin; rather, it was originated artificially in the direction of Kendall Square. The presence of this spike did not significantly affect our determination of the lowest fre- quency of the measured distribution, though it increased the systematic error in the determination of the highest redshift of some of our measurement sets. Calibration of the telescope is accomplished through the use of a noise diode whose noise temperature is pre- set at Tnoise ≈ 115 K. The telescope begins calibration by measuring the average power it receives when the noise
  3. 3. 3 diode is on, pwr1, the average power it receives when the noise diode is off pwr0 and then calculating the following temperatures [? ]: Trcvr = Tnoise + pwr0 pwr1 − pwr0 − Tspill (3) Tsys = Trcvr + Tspill (4) Here Tspill = 20 K is the amount of radio power in Kelvin that the dish fails to reflect into the antenna feed. When measuring the power, the SRT software processes the power signal in various ways. At the final step of pro- cessing the signal is multiplied by a quantity referred to as calcons to yield the final result for the measured power. The calibration process conclues when the SRT uses the values it calculated for (3) and (4) to change the scale for temperature by setting: calcons = Tsys × previous calcons pwr0 (5) The default setting for calcons if no calibrations have been made is 1.0. IV. PROCEDURE AND DATA ANALYSIS FIG. 4. Detail of the top of the plot from Figure 3, which is the raw spectrum for a 600-second exposure at l = 55. The peak of the distribution is slightly redshifted away from 1420.4 MHz. The highest redshift in the figure was determined to be about 1420.3 MHz. We preformed a 600-second exposure for each galac- tic longitude from 5 to 90 in steps of 5. The detail of a measured spectrum is shown in Figure 4. We see that the larger part of the distribution in this case is generally redshifted, suggesting that the galactic matter closest to the Sun in the direction of l = 55 is moving away from us. Note also that the redshifted distribution is not a peak, but rather a broad distribution of frequencies. This is predominantly due to the fact that when the antenna is pointed towards l = 55 its beam crosses multiple or- bits concentric to the Sun’s, whose corresponding velocity projections towards the Earth v(R)sin(δ) have different values. Formula (1) and our theoretical discussion predict a maximum value for ∆vb and thus for the observed red- shift, so any power measured in the far left of the spec- trum should be exclusively due to noise. Thus we av- eraged the power in this region to determine the mean noise level, and then determined the leftmost frequency of the central distribution with power above this level to be the highest redshifted frequency. This procedure was repeated for each spectrum we measured to deter- mine fmeas(l). The resulting values for v(R) are shown in Figure 5. The error bars were calculated based on the estimated systematic error associated with the determination of the maximum redshifted frequency, the uncertainty of the Sun’s distance from the galactic center, and the statisti- cal error involved in the calculation of the last two terms of equation (1) by the SRT software. These errors are summarized in Table 1. The uncertainty in v(R) was pre- dominantly due to the systematic error in the measured frequency σfreq—for example, for l = 5 the 0.1MHz er- ror in the frequency corresponds to a 10.55 km/s error for v(R) so it is very clear in this case the other errors were quite neglible. FIG. 5. Orbital velocity of matter (atomic hydrogen) as a function of its distance from the center of the galaxy. We took the vLRS to be equal to 220 km/s. A recent paper [5] determined the rotation curve of the galaxy by following different tracer objects in it. The pa- per contains a plot of the measured values of the velocity curve given the assumption that vLSR = 220 km/s and R = 8.5. For R > 3 kpc the values recorded in this plot are within 1-2σ of our data points, lending credence to our measured values in this range. Closer to the center of the galaxy the difference between our measured values and the papers grows, likely due to the limited resolu- tion of our equipment and the relatively short time of our exposures. If we take the distribution of matter of the galaxy to be spherically symmetric then Gauss’s Law, when combined with the centripetal force equation, implies that:
  4. 4. 4 l σfmax σv +vEarth σR v(R) ± σv(R) 5 0.1 0.03 0.06 67.09 ± 10.55 10 0.06 0.02 0.12 170.71 ± 21.11 15 0.06 0.02 0.18 185.89 ± 12.67 20 0.06 0.02 0.24 204.76 ± 12.68 25 0.05 0.02 0.30 216.78 ± 12.68 30 0.04 0.02 0.35 228.16 ± 10.59 35 0.04 0.02 0.40 234.58 ± 8.51 40 0.04 0.02 0.45 242.07 ± 8.52 45 0.05 0.02 0.49 244.60 ± 8.54 50 0.03 0.02 0.54 247.88 ± 10.64 55 0.04 0.02 0.57 247.79 ± 6.50 60 0.04 0.03 0.61 237.98 ± 8.59 65 0.03 0.03 0.63 239.57 ± 8.60 70 0.02 0.03 0.66 233.23 ± 6.56 75 0.03 0.02 0.67 235.85 ± 4.56 80 0.02 0.03 0.69 238.78 ± 6.58 85 0.03 0.02 0.70 238.21 ± 4.59 90 0.02 0.03 0.70 235.98 ± 6.58 TABLE I. This table summarizes the errors that were propa- gated in order to determine the error in v(R). The second col- umn gives the systematic error of frequency measured in MHz, the third column gives the statistical error in the velocities cal- culated by the SRT software in km/s, the forth column gives the uncertainty in R due to uncertainty in R = 8.33 ± 0.35 kpc, and the last column gives the values of v(R) with the final propagated error. M(R) = Rv(R)2 G (6) Figure 6 shows a plot of the mass as a function of the ra- dius for this model. Note that it predicts that the sphere of radius 8.33 kpc centered about the galaxy should have a mass content of 1.08 ×1011 M . The sum of the masses of the bulge and disk of the galaxy is approximately 8 × 1010 M [6]. If we assume that the galaxy has the distribution of matter that is visible to us through elec- tromagnetic radiation the spherical model we have as- sumed will not give the exact value for M(R ), but it should give a good order of magnitude estimate. This being the case it seems we have just shown that there is more mass towards the center of the galaxy than the whole galaxy contains. Clearly this poses an issue for our understanding of the galaxy and gravity. The most popular proposed solution to this discrepancy is the idea that the galaxy contains a large amount of ”dark” matter that interacts weakly with light, which would explain it has never been observed directly [7]. The best current understanding is that there is a halo of dark matter around our galaxy, and this halo contains about 90 percent of the total mass of the galaxy [8]. Since the halo is spherically symmetric and contains FIG. 6. A plot of the amount of matter contained within radius R based on a spherically symmetric model of matter distribution. The error bars were calculated based on the uncertainty of v(R) and R. most of the mass of the galaxy Figure 6 should in fact be a reasonable approximation to M(R) for the Milky Way. V. CONCLUSIONS We exploited the properties of the 1420.40 MHz line emitted by atomic hydrogen to determine the orbital speeds of matter around the galaxy at different distances from its center. The data we found agreed well with recently published data for values of R > 3 kpc. By deriving the amount of matter that should be contained within the orbit of the Solar System based on the veloc- ity curve for the Milky Way we concluded the amount of matter visible in the galaxy is not sufficient to account for the fast motion of galactic objects. This conclusion conforms to the current theory that the galaxy is predom- inantly composed of dark matter, which is the ”missing mass” whose gravitational pull causes objects to move faster around the galaxy than could be explained by the presence of the matter we can see alone.
  5. 5. 5 [1] F. H. Shu, The Physical Universe: An Introduction to Astronomy (University Science Books, 1982). [2] “Jansky, Karl (1905-1950),” (2007). [3] 21 cm Radio Astrophysics, MIT Department of Physics. [4] J. D. Kraus, Radio Astronomy (Cygnus-Quasar Books, 1986). [5] P. Bhattacharjee, S. Chaudhury, and S. Kundu, arXiv:1310.2659v2 [astro-ph.GA] (2013). [6] J. E. Barnes, “Populations and Components of the Milky Way,” Accessed on 2-11-2013. [7] V. Trimble, Annual Review of Astronomy and Astro- physics (1987). [8] Battaglia et. al., Monthly Notices of the Royal Astronom- ical Society (2005).