UC Berkeley Astrophysics 121 Radio Lab Professor Bower HI Mapping of the Spiral Arms of the Milky Way

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1 UC Berkeley Astrophysics 121 Radio Lab Professor Bower HI Mapping of the Spiral Arms of the Milky Way Adam Fries (Dated: May 12, 2009) The 21-cm line refers to the spectral line created by changes in the energy state of hydrogen and occurs at approximately MHz. Because this line appears in the microwave, it can easily penetrate dust and interstellar smoke which plagued optical astronomers for centuries. Because the universe in extensively comprised of neutral hydrogen, observing it at this frequency becomes a superior method in mapping the structure of large scale objects. In this experiment we observed hydrogen along the great plane of the galaxy by recording the observed spectra. This data was corrected to the Local Standard of Rest and comparisons between the extracted properties were made. In result, a region of hydrogen in the galaxy was sufficiently mapped along with an estimation of the associative rotation curve and estimation of the mass distribution up to R 0. I. INTRODUCTION The HI Mapping experiment employs many of the skills developed thus far in the laboratory. Not only is an accurate conversion between coordinate systems essential, but the techniques in reducing data such as Fourier filtering become fundamental in producing reasonable results. The first recorded person ever to attempt to optically map the structure of the Milky Way Galaxy was Sir William Hershel. In the late 1700s his attempts at doing so led him to conclude that the our galaxy was in the shape of a disk. Unable to appreciate the clouding effects of interstellar dust, Hershel s model of the galaxy resembles little of what it observed today including a placement of the Sun at its center. In following, we investigate the observed temperature of the gas composing our galaxy. II. THEORY A. Neutral Hydrogen and the 21-cm Line Much of the matter of galaxies is composed almost entirely of enormous clouds of hydrogen gas. Accompanying all this hydrogen is the Cosmic Microwave Background radiation. Because the radiation of the CMB is energetic enough to excite the hydrogen, we can expect these higher energy atoms to interact and collide into one another creating a population inversion into these higher states. As a result, a photon with a frequency characteristic of the energy level difference induced by these collisions is emitted once the atoms decay back down to the unexcited state. Detection of photons characterized by this frequency, therefore implies a detection of hydrogen in the universe. The emission of this characteristic photon occurs from the following process. First, both the proton and the electron constitute the respective magnetic dipole moments: µ p = g pe 2m p S p, µ e = e m e S e, where g p 5.59 (1) Subsequently, any magnetic dipole will generate a magnetic field such that B = µ 0 4πr 3 [3( µ ˆr)ˆr µ] + 2µ 0 3 µδ3 (r), (2) where δ is the delta distribution. Now, the Hamiltonian for our electron in the presence of the magnetic field produced by the proton s dipole moment is then H hf = µ e Bp (3) The result from first-order perturbation theory states that the first-order correction to the energy is the expectation value of the perturbing Hamiltonian: E 1 n = ψ 0 n H hf ψ 0 n (4) Therefore, we arrive at the following energy state of the perturbed atom. E hf = µ 0g p e 2 3πm p m e a 3 S p S e (5) The dot product of spins in Eq.5 describes the spin-spin coupling between the interaction of the proton and electron. This produces a triplet energy state of hydrogen where the spins are pointing in the same direction, and a singlet state where they are opposite. The difference in these energy states are described by the following (Griffiths 1995): E 1 hf = 4g p 4 3m p m 2 e c2 a 4 E = { +1/4, (triplet) 3/4, (singlet) 4g p 4 3m p m 2 ec 2 a 4 (6) Once the hydrogen atom is excited to the triplet state, it decays back down to the singlet state and emits a photon

2 B Galactic Mechanics Adam Fries with frequency corresponding to this energy level difference. By today s best estimates (Wohl ress), the corresponding frequency of emission is ν = E h 1, 420, 405, ± Hz (7) Because the thermal energy of the Cosmic Microwave Background radiation (k b T = 2.3(10 4 ) ev) is a few orders of magnitude greater than E ( 5.9(10 6 ) ev), we can expect a great deal of these energy level transitions in hydrogen and therefore detect an abundance of these frequency emissions. B. Galactic Mechanics 1. Line of Sight Velocities The Milky Way Galaxy is a complex, dynamic and evolving object that continues to bear new insights as to its structure as more resources and efforts are lent to it. However, for the general purposes of this experiment, we are allowed to assume that it rotates simply as a rigid body with a constant angular momentum for our measurements. With this in mind, our goal is to find the distance of detected hydrogen to the Galactic center in order to map its position with respect to the center of the Milky Way. First, we let R be the position vector of detected material with respect to the Galactic center, and let the angular velocity of rotation be Ω(R). The velocity of the hydrogen gas in the galaxy is then v = Ω R (8) When observed from our position defined by the Local Standard of Rest when Doppler corrected, our velocity is defined as v 0 = Ω(R 0 ) R (9) where R 0 is defined as the radius of the solar circle. The subsequent line of sight velocity (from our perspective) of this gas is the projection of the velocity difference v v 0 on to the vector (R R 0 ) which runs from the Sun to the gas of interest, such that υ los = R R 0 R R 0 [Ω(R) R Ω(R 0) R 0 ] (10) Eq.10 can be simplified by employing a few vector identities: a (a b) = 0 and a (b c) = b (b a). As a result, we arrive at υ los = [Ω(R) Ω(R 0)] (R 0 R) R R 0 (11) Additionally, R 0 R = R 0 sin θˆn, where ˆn points in the perpendicular direction of the galactic disk. Depending on the orientation or direction of galactic rotation, this quantity can be positive or negative. Likewise, from the law sines sin θ/ R R 0 = sin l/r. And because Ω(R) = V (R)/R, we can therefore plug these statements into Eq.12 to arrive at υ los = ( V (R) R V (R 0) R 0 ) R 0 sinl (12) Thus, If we observe any part of the galaxy and imagine that this part is associated to a ring of radius R, then after Doppler corrections, we can only expect to distinguish only one line of sight velocity associated with that position and that this velocity is proportional to sinl (Binner & Merrifield 1998). 2. Dark Matter and the Keplerian Decrease From working with the planetary observations made by Tycho Brahe, Johannes Kepler formulated his 3 laws of planetary motion which later became motivations for Newton s 3 laws. In particular, Kepler s 3rd law states, the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. Mathematically this can be represented in the following way: P 2 1 a 3 1 = P 2 2 a 3 2 P is the orbital period and a is the semi-major axis (13) Eq.13 implies that given a two bodies, 1 and 2, that if body 1 is farther away from an origin than body 2, then body 1 traverses a distance greater than body 2. Moreover, this relationship implies that their respective orbital velocities are inversely proportional to their radial distances from an origin. Furthermore, this is a similar statement to that of Eq.8 where Ω is held fixed for a rigid body. As the orbital radius increases, then accordingly, the orbital velocity should decrease. In 1970, Vera Rubin and Kent Ford looked at emission lines from regions of hot ionized gas in the Andromeda Galaxy and were able to estimate orbital speeds out to a radius of about 24 kpc. However, their results showed no sign of Keplerian decrease in the orbital velocity of the galaxy as the radius increased (Ryden 2003). Since the detected light from baryonic matter in the galaxy corresponded to a lower than observed orbital speed, the presence of some missing mass was presumed. This missing mass is called dark matter and accounts for the somewhat constant orbital velocity past a radius expected to exhibit the Keplerian decrease. Observationally, Kepler s 3rd law approximately holds true up to a certain galactic radius when observing large scale structures such as galaxies. If we are only concerned with gas outside the solar circle, then we can make an assumption about the orbital velocity of the gas motivated by the presence of dark matter. In particular, we 2

3 C Specific Intensity and the Brightness Temperature Adam Fries set V (R > R 0 ) = V (R 0 ) 220 km s 1. Consequently, Eq.12 now contains only one unknown R, the radius to our gas from the Galactic center with respect to our observed line of sight velocity. R 1 = υ los V 0 R 0 sinl + R 1 0 (14) Subsequently, the orbital velocity defined in Eq.12 as V (R) can be related to the total mass the of our galaxy from the following V (R) 2 = GM galaxy R 1 (15) C. Specific Intensity and the Brightness Temperature If we observe an object, we do so by recording the number of photons with a detector. But this measurement alone introduces ambiguity so it becomes relevant to introduce a time interval between collections along with a area over which we are detecting the source. This measurement is known as the intensity of the object and is in units of energy per unit time per unit area. However, in order to obtain a general quantity with which we can characterize a source of light, we need to introduce and combine with the intensity the size, and direction of the source of interest (Dept. 2009); moreover, the specific intensity, where I(ν) = 2hν3 c 2 1 e hν/k bt 1, where hν/k bt 1 (16) which in the classical limit reduces to and has units of I(ν) = 2k bt λ 2 (17) [I(ν)] = ergs s 1 cm 2 Hz 1 ster 1 (18) Eq.18 informs us how much power we can expect from a source provided a solid angle, thus we can view the specific intensity of a source as its angular surface brightness. One important result from defining specific intensity to consider while observing a source is that I(ν) is constant along a ray path. That is, it is independent of distance. This can be shown by considering the energy seen by a pixel from a detector. This differential energy is given as de = I(ν)dA dν dt dω (19) The subsequent power is then just P = de/dt. If we consider observing the Sun, then this power is then P = I (πr 2 lens)ω (20) Here, we set out to show that the intensity of the focal plane of the detector or lens is the same as the intensity at the surface of the Sun. The intensity at the focal plane is given as the following I lens = P A Ω lens = I (πr 2 lens )Ω A Ω lens (21) Furthermore, the area of the Sun on the focal plane is defined by A and the solid angle of the lens is given as Ω lens and are given as A = πr 2 d 2 D 2 Ω lens = πr2 lens d 2 (22) where d is the focal length and D is the distance between the detector and the Sun. Plugging this back into Eq.21 shows that indeed, the specific intensity is equivalent. I lens = I (23) In practice, it is more convenient to think about specific intensity in terms of temperatures. Rearranging Eq.17 allows us to view the angular surface brightness, or more simply, the brightness temperature. T B = I(ν) 2k b λ 2 (24) However, in practice T B is not equal to the physical temperature of the object. Nevertheless, our observed object emits an intensity and is characterized by the brightness temperature that produces it (Dept. 2009). By applying a similar technique of comparing intensities as before, we can relate the brightness temperature with the antenna temperature. Because 2k b T A (ν) = P(ν) = 2k b T B (ν) (25) then again we have an analog expression to the related intensities such that T A (ν) = T B (ν) (26) but only if the solid angle of our source is larger than the antenna beam width. This is not assumed to be true for our experiment. Thus, we appeal to the following general relationship: ( ) Ω source T A (ν) T B (27) Ω source + Ω beam III. EXPERIMENTAL EQUIPMENT AND PROCEDURE A. Signal Path at the Leuscher Facility In this experiment, we employ the use of the Leuschner 4.5 m radio antenna. This antenna site is located at the 3

4 A Signal Path at the Leuscher Facility Adam Fries Russell Reservation in Lafayette, California and rests at an altitude of 304 m. According to Fig.1, the source signal corresponding to the 21-cm line it receives is first amplified before being filtered by MHz bandpass filter which allows only signals whose frequencies are close to that of hydrogen to enter the system. The signal is then mixed with the first local oscillator (ν lo1 = MHz) and filtered through a MHz bandpass filter and once again amplified. All of these processes occur at the dish and the resultant signal is sent through a long coaxial cable to the Control Room at Leuschner to be further process. Because the signal travels a significant distance through some impedance in the cable along its way to the Control Room, it must be amplified once again. At the Control Room, the source encounters what is reasonably half of an SSB mixer where it is first split and each constituent mixed with a phase-shifted local oscillator (ν lo2 = 150 MHz). Each resultant signal encounters a low pass filter and is read by the ADC unit. B. The IBOB and Real-time Processes In conjunction with the Leuschner antenna, the sky signal is read and processed by an IBOB-based (In Board Out Board) spectrometer. Designed by Andrew Siemion, the spectrometer has been optimized for narrow band, high resolution spectroscopy that performs at a sampling rate of 24 MHz with 8192 samples which yields a bandwidth of 12 MHz (Andrew Siemion 2009). The IBOB is designed specifically to encounter a signal whose frequency is approximately 150 MHz, much higher than the afore mentioned sampling frequency, by taking advantage of the aliased signal this method introduces. This design allows the system to operate and be built with the most cost effective, highly efficient components. The advantage of the CASPER designed IBOB is that the data is constantly being read and converted to digital in real time. This means that as the sky signal is received, the data is converted, integrated and transformed into the spectral domain continuously and nearly instantaneously. To get an idea of the IBOB s speed and efficiency, let s imagine that for our experiment, we want to average over 2000 spectra. As mentioned previously, the sampling frequency and number of samples per spectra are known. By multiplying these quantities, the total integration time per spectra can be found from the following: [( 24(10 6 ) samples s ) ] 1 1 spectra samples Then, by considering 2000 spectra, 341 µs spectra (28) µs (2000 spectra) 0.68 s (29) spectra The total integration time here is less than a second, thus making the overall processing time impressively speedy and essential for acquiring large amounts of data. IV. RESULTS FIG. 1: The signal path of a radio source collected at the Leuschner facility is illustrated above. This block diagram includes paths for both HI and OH. However, for this experiment we only consider the path corresponding to an HI source. The central theme behind this experiment was to record the antenna temperature corresponding to hydrogen gas for a particular region of the Milky Way Galaxy, namely along the Galactic plane for values of l from 10 to 250 and b = 0. We successfully acquired data for a range of l between 4 and 220 which was enough to adequately express temperature trends in this region in order to draw conclusions. The spectra from Fig.2 is a result of averaging overing 256 spectra collected at (l, b) = (24, 0 ). After filtering the signal we obtain a sense of our reduced signal to noise which approaches a very large number at this point. From this data, four comparisons were made: (i) as illustrated in Fig.3, a plot of galactic longitude versus the observed line of sight velocity was made, (ii) a mapping of observed temperatures was made for an area of 625 4

5 Adam Fries and flipped in order to match. Here, we notice that the two plots somewhat coincide and thus believe our claim that we are actually mapping spiral arm structure. By observing the galaxy at some galactic longitude, we can expect to observe the hydrogen gas moving at a range of orbital velocities. Within this group a subsequent maximum velocity exists. By definition, υ max belongs to a tangential radius projected from the center of the galaxy. Therefore, every angle of observation contains a gas moving at some υ max belonging to some Galactic Radius. Fig.6 represents this motivation and indicates the Milky Way s rotation curve for gas within the solar circle. The plot illustrates that the velocity of the system increases with radius but begins to settle around 200 km/s around 8 kpc and reflects the following model of motion of Eq.15 here: FIG. 2: The spectra above, is an example of a typical sample of our source of interest. This particular spectra has many features associated with. We assume a range of velocities here where the power rises sharply above zero. Each point represents a particular velocity and subsequent radius from Eq.12. Likewise, each radius exhibits a unique temperature. sq. kpc in Fig.4, (iii) the galactic rotation curve for the observed tangential velocities of the Milky Way inside the solar circle for (0 < l < 90 ), (iv) as well as the associative galactic mass versus radius. The l υ curve in Fig.3 represents the changing velocities observed with changing galactic longitude. The observed temperature gradient of this plot implies that a significant portion of gas is moving toward our position when observing at angles greater than l = 90. While for angles less than l = 90, the opposite is true. In general, we can extract an overall tendency toward a particular direction of rotation for the Milky Way. The main focus of this experiment was to map hydrogen in the afore mentioned region of the Milky Way. According to Fig.4, we notice a very distinct structure associated with observed temperature variance. Because the shape we observe of the hotter gas resembles that associated with spiral arm structure, we checked the validity of such a statement with a previous mapping of gas thickness conducted by Levine et al., Fig.5. Fig.5 represents detailed map of the hydrogen gas thickness which refers to the structure of the spiral arms. In comparison with Fig.5, our observed map is rotated π/2 to the left υ 2 max = GMR 1 (30) Consequently, the mass of the Milky Way as a function of radius can be easily deduced from Eq.30. This distribution is shown in Fig.7 and for out to 8.5 kpc, the mass is estimated to be approximately 10 million solar masses. By today s estimates the entire mass of the Milky Way is closer to 10 billion solar masses. Given the existence of dark matter, and that the dark matter halo ispredicted to be spherical and extends beyond the arms of a galaxy, we may assume that our low estimates for Galactic mass may be justified. V. CONCLUSION The main goal of this experiment was to observe and map the behavior of hydrogen in the Milky Way for a particular range of angles. In doing so, we were able to make comparisons between observed line of sight velocities and how they change with angle in order to provide a motivation for the overall direction of rotation of the galaxy. In addition, we mapping the temperature distribution of hydrogen gas and found it to exhibit structure resembling a spiral arm. From these comparisons, we were further able to plot the galaxy s inner rotation curve assuming solid body dynamics as well as estimate the mass there within. The results of this experiment reflect careful planning and accurate projection of the observed object as well as patience in the data taking process. Andrew Siemion, C. t. 2009, Design Gallery, Spectrometers, Leuschner Spectrometer Binner, J., & Merrifield, M. 1998, Galactic Astronomy (Princeton University Press) Dept., U. B. A. 2009, UC Berkeley Astrophysics Radio Lab Experiments Griffiths, D. 1995, Introduction to Quantum Mechanics (2nd ed.) (Pearson Prentice Hall) Levine, E. S., Blitz, L., & Heiles, C. 2006, Science, 312, 1773 Ryden, B. 2003, Introduction to Cosmology (Addison Wesley) Wohl, C. G. in progress, More than one Angular Momentum 5

6 Adam Fries FIG. 3: The Doppler corrected line of sight velocities are plotted above against galactic longitude. Here we notice an implied density of hydrogen along the zero velocity line for all angles. Because gas on the Solar circle moves at the same velocity as the Sun, we would not expect to observe a difference in the velocities. Therefore, this line represents the observed gas on the solar circle. 6

7 Adam Fries FIG. 4: The plot above indicates a temperature gradient as a result of mapping hydrogen in the indicated regions. The regions of higher antenna temperature imply regions with a greater density of hydrogen. This leads us to believe that the observed hot gas represents a spiral arm of the Milky Way Galaxy. FIG. 5: The plot above is a result of the work by Levine et al. documenting the perturbations in gas thickness of the Milky Way. The grayscale regions represent the highest thickness in gas when compared to the local medium (Levine et al. 2006). Here, we use this plot as a standard map to check with our results of Fig.4 7

8 Adam Fries FIG. 6: The observed rotation curve above indicates a reasonable fit for our prediction of υ R 1/2. The plot indicates a large density of points at approximately 8.5 kpc which implies that the orbital velocity is approaching an asymptote and thus reaching a maximum. In order to verify this assumption, the rotation curve applied to radii greater than R 0 is required. FIG. 7: Assuming that the Milky Way Galaxy acts as a rigid body inside the solar circle, the plot above shows the mass of the galaxy as a function of radius. At approximately 8.5 kpc, a mass of kg is estimated. However, outside the solar circle it is well established that the Milky Way does not act as a rigid body in that its angular velocity is no longer constant. Therefore, we expect not to believe our estimate of galactic mass here. Moreover, the observed estimation of the mass is not nearly enough to account for our assume lack of Keplerian dropoff. 8