The Total Mass And Size Of Our Galaxy

THE TOTAL MASS AND SIZE OF OUR GALAXY ROBERTO BARTALI

ABSTRACT The knowledge of the size and mass of our Galaxy, called The Milky Way (MW), is a difficult problem, but it is very important because is our home in the Universe. Measuring and understanding our Galaxy, can tell us how it formed and it will be easy to do a generalization for the other millions of galaxies in the Universe. Really, we know much more of very far galaxies than of the one we live in, this is not a surprise because we are inside the Milky Way and very close to its equatorial plane, due to this fact, we can only see stars above and below us, but not those in the other side, so the first problem to be solved is to find the form and the size of the MW, then, the position of the Sun in it. After the discovery of the limits of the Galaxy, we can measure the mass, with indirect and statistical methods, because we can not count all the stars and nebulae it has. The reader will be find in this work how astronomers try to find the geometry and dynamics of the MW, I say “try” because there are not a definitive model for that. INTRODUCTION The MW (figure 1), as seen by naked eyes, is a beautiful “long cloud” that cross all the sky, very impressive in the southern hemisphere. For that reason, ancient astronomers called it a river or a milky flow, the name mean precisely that: “path of milk”. The first attempt to explain what was that cloud, was made by the greek astronomer Democritus five centuries before Christ, he Figure 1 think it was made by a lot of stars. He was The milky Way galaxy, full sky view, and two right, but for more than 21 centuries no one satellite galaxies, the Magellanic Clouds can demonstrate that. Galileo Galilei was the first to point a telescope to the MW and he discovered that there are an enormous quantity of stars, too closer for the eyes to be resolved. Later, W. Herschel, with much more powerful telescopes (figure 2), make an entire map of the MW and discover that it is a squeezed bubble with an height 5 times less than the length (figure 3). He concluded 3 things: the Sun was in the center of the MW, the Sun is moving respect to the stars and the density of the stars was always the same Figure 2 everywhere. One of the Herschel’s No much improvement of the knowledge was made until telescopes 1

the XX century, when Schwarzschild, Kapteyn and Seeliger developed statistical methods to solve the problem, later, Seares, among others, improve and redefined some of their assumptions. Observing toward the center of the MW, and generally through the spiral arms is almost impossible. Just to a distance of a few thousand parsecs, the absorption, due to the interstellar gas and dust, may reach many magnitudes (sometimes 1 Figure 3 magnitude every 1000 parsec), so the only way to see the other Map of the Milky Way side of the Galaxy is with a radiotelescope, technology made by Herschel available only from just some 40 years ago. Observing the 21 cm wavelength emitted by neutral hydrogen (figure 4), we finally confirm that our Galaxy really is a spiral galaxy of some 100 Kly across, with the Sun in an arm at a distance of 8 Kpc from the center. In the last few years, observatories in space give a much detailed information about the size and the mass of the MW, because they observe at many different wavelength. Most of these observations are impossible from the Earth surface due to the atmospheric absorption and refraction. The analysis of the new data and precise calculation of the movement of the stars in different part of the MW, instead to give answer, raised more problems because there are much many matter than that we can see. In the following sections I will try to give the better Figure 4 figure of the mass and size of our Galaxy explaining the Early radio map of the Milky Way spiral arms methods used to obtain it. MILKY WAY GEOMETRY DETERMINATION In the first half of the XX century, many astronomers try to define, with many methods, the right size and form of the MW. They really do a great job because the observational data and computing power they have was very poor, so it is not a Figure 5 surprise if their result diverge substantially from our Kapteyn conception of the MW actual knowledge. J.C. Kaptein obtained a model of the MW (3)(10), using statistics, taking into account only the most luminous stars of less than 12th magnitude in selected areas. In his model, he considered the Sun in the center of the Galaxy, resulting in 30Kly length and 4Kly height (figure 5), all the stars was distributed in concentric circles, rotating around the center. This model resemble only partially the real MW, he failed considering the Sun in the center and also he never take into account the effect of the light absorption by dust. Later, F.H. Seares, using the same Kapteyn selected areas, but with a limit magnitude of 18, discovered that the Galaxy is bigger. Measuring the different stellar density in different galactic longitudes, he also find that the center of the MW is not coincident with the Sun, but it is in the direction of the Sagittarius constellation, where is also the center of the globular cluster system. Seares also do a better analysis of the stellar

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density, this way he find that the Sun is in a peripheral position, because the density increase toward the center and decrease toward the outer side of the MW, the accuracy of the Sun position respect to the Galaxy center is, as he said, in strong dependence with the value of the interstellar absorption. Their results, are more important in the sense that they contributed to the next studies of the Galaxy geometry, than for the figures they find. Around 1960, the stellar density structure of the MW was well known, brightest and more massive young stars, in the galactic plane, then, going toward the poles Figure 6 less massive and late spectral type stars arranged in O, B stars and gas clouds in galaxy arms (NGC2997) layers until 2Kpc where the density is only 1/100 of the galactic plane one, in those years, astronomers think that they reach the border of the galaxy at that distance. This analysis find an asymmetry, so the Sun is clearly displaced from the galactic equatorial plane to the north some 14 pc (from 10 to 20 pc depending on the author, we do not know the exact value). Another interesting thing about stellar density analysis was that it revealed a great increment close to the center, what we know now as the bulge. Observing in the direction of the MW Figure 7 center, at infrared and The Milky Way and the Globular Cluster radio wavelength, some distribution (empty circles) astronomers (1)(6) have determined that the central bulge is not flattened spherical, but it is barred, as G. de Vancouleurs said 30 years before. When astronomers extended the analysis in all directions, they find that the resulting density was clearly not uniform, but they do not know the spiral arm structure, so they find no relationship between spectral class and distance, looking at certain galactic longitudes they find increment of young stars, than another increment at a greater distance; for example, they find that OB stars (figure 6) are condensed is groups with dark and bright clouds between them and only close to the equatorial plane until about 10 degree of galactic latitude (we know that the spiral arms contains dense areas of OB star associations and also there are large and dense clouds). Their conclusion was Figure 8 that the Galaxy is a very complicate system and The Milky Way observed in different the analysis bring fidelity only locally, each part wavelength from space and by advanced of the MW has to be treated alone. technology telescopes

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Stebbins, Whitford and others, tried to solve the problem with the analysis and observation of Globular Clusters, but Shapley was really who did the best job. Observing a large quantity (near 93) of Globular Clusters, he find many Cepheid type variable stars, the strong relationship between the luminosity and the period, permit to him to compute their distances accurately and to define their relative position respect to the Sun (figure 7) showing that they form a near spherical halo which diameter is almost as large as the disk one. But, due to the wrong account for absorption, he estimated wrong distances. Shapley work is one of the best, because it put, also, the basis of the understanding of other galaxies and for the dynamic studies, enlarging the MW gravity influence much more over the disk. He shows that definitively, the center of the MW is in the direction of Sagittarius and the system of the Globular Clusters orbits the Galaxy around that same point. Now, in the Space Era, many technological and computational improvement, like very large telescopes and orbiting observatories, have revealed a better view of the MW, essentially we have a more precise and high resolution data, but the size and the shape are not changed considerably, only best figured.. Spiral arms are composed of HII regions, molecular clouds and OB star associations, older stars, are observed at higher latitudes above and below the galactic plane. The better way to trace the arms is observing at different wavelength, (figure 8), the star forming regions and going to the limit of sensitivity, is possible to delimitate each arm, because in the visible band we can only see the young bright stars and the gas that they excite, but other wavelength shows the inter arm stars and matter. Precise measurement of the H-alpha line in the spectra of these regions (2) and better accuracy in the velocity of stars, revealed the real structure and shows that the MW is a Grand Design Spiral (figure 9 and 17), but, up to time, the discussion is around the number of the spiral arms, which some authors detect three and others can count four. Figure 9 Grand Design structure (four spiral arms)

MILKY WAY MASS DETERMINATION

Figure 10 Milky Way velocity profile

The Milky Way is a rotating system, stars, gas and even the central black hole (13) are moving in different way depending on the position they are occupying respect to the galactic center. Basically, the velocity of each object depends on the total mass of the galaxy and of the object mass, because the object feels a gravitational force that is proportional to the masses involved, but obviously, astronomers can not measure the mass in direct form. The first approach to the determination of the mass, is to measure the velocity of the stars at different distances from

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the center of the MW, as far as 1920-1930 Oort shows that the rotation is not a linear function of the radius, but it is differential (figure 10). Regions close to the bulge rotate as a solid disk with a constant angular velocity, but at a distance of about 1 Kpc the velocity profile change, another change occurs at a distance of about 5 Kpc (the Solar System is at 8 Kpc from the center), and until some 10Kpc it follows the Kepler third law (eq. 1), then, at larger radius, velocity increase with the distance. To explain this fact, we need a lot of mass to increase the gravitational attraction, but the mass of the disk decrease with distance, the same is true for the galactic halo. The implication of this fact is that the most of the mass must be in the halo, but the density, taking into account stars, globular clusters and gas (comprising almost all of the visible matter at any wavelength), is very low, much less than the disk one, so something not visible but, in a very large proportion, must be there. Astronomers calls it “Dark Matter” and whatever it is, it must be a lot because the ratio of luminous to dark mass is about 1/10, so 90% of the Galaxy is dark, otherwise there is no logical and physical reason to explain the velocity curve. To measure precisely the rotation of the MW is very difficult, because we are inside it and we are moving also around the center at a considerable speed. The third Kepler law is useful to calculate a mass, but it need the orbit radius, and this is a quantity not known for sure. Shapley made a great contribution to find a solution, because he studied the Globular Cluster system, which members are not orbiting in the plane of the MW, but on planes randomly distributed, thus their velocity respect to us are very high, therefore easiest to measure. The Sun is moving in a near circular orbit around the center of the MW at an average speed of 220 Km/s but it is not a constant movement, it is accelerating in three different ways (10..11 Km/s faster toward the galactic center, 5..7 Km/s faster in the direction of the rotation and 7..8 Km/s toward the north galactic pole), this implies more corrections to apply when observing other stars and try to measure their absolute velocity, we need to define a reference coordinate called “Local Standard of Rest” (LSR) that take into account for that as best as possible. Only after the elimination of the Sun movement from the stars apparent movements, we find their real motion; now, using the Newton’s form of Kepler third law: 4(pi squred)(a cubed) (eq. 1) sqared(p) = -------------------------G(M1+M2) Where p is the orbital period, a is the semimajor axis, G is the gravitational constant, M1 is the Galaxy mass and M2 is the star mass. But the star mass (M2) is very low compared with the Galaxy mass (M1), so M2 can be eliminated, if we also suppose that the orbit of the star is nearly circular, then the semimajor axis is equal to r (circle radius), next we know that 2 (pi) r (eq. 2) P = -------v Where p is the orbital period, r is the orbital radius in parsecs and v is the orbital speed in Km/s; introducing equation 2 into the equation 1 and reducing we get r (v squared) (eq. 3) M1 = ----------------G 5

this equation shows the mass (in solar masses units) inside an orbit of some radius. Equation 3 works well only for a small portion of the MW disk, as we see in the figure 10. At large distances, outside the Sun’s orbit, the calculated galactic mass, in this simple manner is not realistic, because M1 will be much greater than the calculated to fit with the observation of the star velocity. The MW mass determination problem will be solved only taking into account Dark Matter, but first of all, astronomers have to find it. The first scientist to proof the existence of the DM in our Galaxy, was D. Schramm. Many theories are issued to explain what it is, but there are no consensus because of the lack of observational material, after all we only see the gravitational effect it exert over the visible matter. The DM seems to be located only in the galactic halo (1)(2), many astronomers try to demonstrate the need of its presence in the disk and the bulge, Oort was one of them, and he found that about 50% of the disk would be DM, but this and other results are well explained with or without the DM (1) depending on the constrains used for calculations. DM can be many things, baryonic or exotic (not baryonic). Figure 11 If we think about baryonic matter, it could Microlensing effect by a dark object be made of planets, low mass and cold stars like brown dwarfs (figure 12), extremely low luminosity white dwarfs, dust, molecular gas, atomic or ionized gas, black holes, neutron stars and normal white dwarfs. After years of observations, only a few objects (4)(5) have been discovered using various techniques, including deep field images of the Hubble Space Telescope and very sensitive and linear CCD detectors on dedicated telescopes, these allows the discovery of brown dwarfs (figure 12). A great effort is issued to the MACHO project (Massive Compact Halo Object). If something massive, but not visible, pass in front of a star or another object very far, the light from that object is bended due to the gravitational lensing effect (figure 11); the light curve of the observed object show an increment in a very special manner. MACHO objects observed in the halo DM (4)(5) are between 0.1 to 0.9 solar masses. All observed baryonic DM can be, at best, only 50% of the DM total mass, the rest must be in the form of exotic particles. We can exclude, from the DM budget, gasses in any form because they emit radiation at some wavelength and, until now there is not observational evidence for that. Also dust is not a realistic form, because we do not see the obscuring effect due to the absorption. The 50% left, and perhaps much more, must be in the form of Exotic DM that came in two forms: hot and cold. Hot DM are low mass particles like neutrinos (and there are many doubt of that, because experiment results are contradictory) traveling very fast, so Figure 12 Brown Dwarf star inside the lines

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very difficult to trace, the lack of charge they have, imply that they only respond to gravity, so even in the extremely large quantity they do not care if they do not have mass. Cold DM are heavier particles that, of course, travel slowly. The technology to detect cold DM is not yet developed, because we first understand and then detect a phenomenon called The Weak Interaction. The knowledge of the extension of the galactic halo is then very important, because all the unseen mass is there. Most investigations are around the study of the tidal effect that the MW exert to a globular cluster or a small satellite galaxy (figure 1 and 13). Each massive object gravitational field is limited in extension by the Roche equation, so the masses of the two objects are related by the Roche criterion as: (eq. 4) r / R = (m / M)e1/3 Where r is the tidal radius of the satellite, m is the mass of the satellite, R is the distance from the Galaxy center and M is the MW mass inside the radius R. Everything that are inside the Roche limit, are gravitationally bounded to the massive object inside it, so the effect that astronomer study is the deformation or the stretches of the intrusion object. In the case of a satellite galaxy, this effect is shown by the destruction of its original form, like the Sagittarius Dwarf (figure 14), stars near the MW galactic center feel a greater attraction than those in the far side, so we can observe a very elongated shape. As gravity is inversely proportional to distance, farter are the satellite galaxy, lesser the tidal effect it experiment, but for gravity, the mass of the involved objects take into account in direct proportional form, so more massive the object, more distant go its gravity attraction. Figure 13 The effect of tidal force in globular Partially map of the MW satellite galaxies clusters is more evident, because they are much less massive than galaxies and they are much closer to the MW center (with some exceptions). We can see it in a deformation of the near spherical shape they have, the orbital velocity increase as they travel inward the center and when they reach certain distance limit (depending on the globular cluster total mass) some stars are separated from the cluster. These stars follows the original orbit of the cluster, but they are no longer bounded to it, as time pass, these stars orbit at longer distance from the parent object, until they are trapped by the MW. If the satellite galaxy has globular clusters, in the best Figure 14 case their orbits are greatly perturbed, in the worst case they Sagittarius dwarf galaxy can be separated and be lost, these clusters belongs now to the MW.

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This method is reliable but it has some limitations that could introduce very large errors in the results. It is very difficult to measure the tidal radius, because if the galaxy is very near to the MW (as Sagittarius dawarf, for example) many stars in the field of view can belongs to our galaxy and not to the satellite (figure 14), so only a strong extrapolation of the observed elongation is not realistic; another thing is that we have to known very well the mass to light ratio for that satellite, but for that, we have to know how much dark matter are inside the satellite galaxy and, of course, we do not. The best approach to compute the mass of the MW is to measure the dynamics of the globular clusters or the satellite galaxies, normally these objects do not have radial velocity because they orbits in a normal plane respect to the Sun plane orbit. The mass can be calculated with the virial theorem in its statistical form: A (eq. 5) M = ------ sum (i=1 to N) of (v squared) r GN i i Where N is the number of objects, v(i) is the radial velocity, r(i) is the distance from the MW center, A is a constant and G is the gravitational constant. Other form to solve the problem is observing and measuring the velocity of objects at larger scale, instead of the close satellite galaxies, the galaxies in the local group. We know that M31 and the MW are moving and approaching one to the other, if we think that this phenomenon is due to the gravitational attraction and this force is greater than the force of the expansion of the Universe, we can find the mass of one galaxy respect to the other. But this have its own problem, because is necessary to know the age of the Universe in order to find for how much time that force was applied and how close the galaxy was before the mutual attraction win over the expansion. Really the solution is very hard to find, in the conclusion the reader can find a summary of the obtained results until now. CONCLUSION Many astronomers try to define, as we have seen, the size and the shape of the MW studying star of different spectral type, planetary nebulae, novae, emission and molecular clouds, variable stars like Cepheid and RR Lyr, and globular clusters. Their results are different, but after 8 decades of intensive observations, from Earth and Space, and mathematical models, there are not enough data to define for sure our Galactic Home. The only thing that nobody discuss is that the MW is a giant Spiral Galaxy (figure 15) and it is the second in size in the Local Group, Figure 15 only surpassed by the Andromeda galaxy M31. But, Spiral arm structure of the Milky Way some people believe in the flattened spherical bulge in the center and others in the barred type center of the Galaxy. Also, the form and size of the Halo is not well accepted, it is suspected a presence of a Corona, outside the Halo with almost twice its diameter. The best accepted geometry of the MW (Figure 16) is a Grand Design Spiral Galaxy (figure 17) as follows: 8

a) Core: 2 AU diameter, very near the radio emitter Sgr A, possibly containing a massive black hole of 2.5x10e6 Solar masses (1)(6)(8). b) Bulge: flattened spherical or barred (1), 20 Kly by 10 Kly, containing very dense neutral hydrogen gas, dust clouds and old stars. c) Disk: Three great spiral arms starting on each side of the bulge, connected by small arms, 100 to 120 Kly by 2 Kly. Contains young stars, open clusters, dense molecular clouds. Evidence of four arm system (2)(6). d) Halo: spherical, about 100 Kly diameter, contain globular clusters, old stars, RR Lyr stars, dark matter, thin clouds of intergalactic matter e) Corona: spherical about 200 Kly diameter (7), contain intergalactic gas, dust at very low density.. From above characteristics, the best accepted classification of the MW is as a spiral galaxy (face on: figure Figure 16 Milky Way structure (side view)

17; edge on: figure 1) Sbc type, but there is a discussion about the not well developed barred nucleus, so the class can be changed for SAB. G. de Vancouleurs classify the MW as a SAB(rs)bcII type including both spiral arms opening and extension and the semi

barred nucleus. Most astronomers agree with a mass of the MW around 10e12 solar masses and with an halo extension, including the DM, of 100 to 200 Kpc of radius. The evaluation of the MW mass is not an easy task, not only because we are in a disadvantage position, but because the variables involved are too difficult to measure and also the error reduction process must take into account for things that are not very well understand. This is why the values in the following table are so different. TABLE 1: Estimation of the Milky Way mass

Year

Author or book

Method

Mass (solar masses)

1976 1969 1973

Astophysical quantities Il Cielo Introductory. Astronomy & Astrophysics Discovering the universe Innanen et al Trimble Fich, Tremaine Zaritsky et al Kulessa, Linden-Bell Peterson, Latham Zaritsky

-

1.4 10e11 1.9 10e12 1.5 10e11

-

2 10e11

Tidal effect Virial theorem Virial Theorem Virial Theorem Virial theorem Virial theorem Hubble expansion

8.9 +/-2.6 10e11 10e12 10e12 8.1 10e11 to 2.1 10 e12 1.3 10e12 5 10e11 13 +/-2 10e12

1990 1983 1987 1991 1989 1993 1989 1989

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Figure 17 Spiral galaxy NGC1232, one of the most similar to the Milky Way

REFERENCES (1): Lopez-Corredoira et al, SEARCHING FOR THE IN-PLANE GALACTIC BAR AND RING IN DENIS, A&A 0104307, 2001 (2): Russeil D, STAR-FORMING COMPLEXES AND THE SPIRAL STRUCTURE OF OUR GALAXY, A&A 2000 (3): Cecchini G, IL CIELO, Utet, 1969 (4): Griest K, ALL ABOUT MACHO, 2000 (5): Glimore G., THE DISTRIBUTION OF DARK MATTER IN THE MILKY WAY GALAXY, A&A 9702081, 1997 (6): Huttemeister S, THE MILKY WAY, STRUCTURE, CONTITUENTS AND EVOLUTION (7): Charlton J., STUDYING THE GASEOUS PHASES OF GALAXIES WITH BACKGROUND QSOs, A&A 0207365, 2002 (8): Backer D, Sramek R., PROPER MOTION OF THE COMPACT, NON THERMAL RADIO SOURCE IN THE GALACTIC CENTER SAGITTARIUS A*, (9): Freedman R., Kaufmann W., UNIVERSE, W.H. Freeman, 2002 (10): Karttunen H. et al, FUNDAMENTAL ASTRONOMY, Springer, 2000 (11): Unsold A, Baschek B, THE NEW COSMOS, Springer, 1991 (12): Sparke L, Gallagher J, GALAXIES IN THE UNIVERSE, Cambridge, 2000

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(13): http://www.eso.org/outreach/press-rel/pr-2003/pr-26-03.html (14): http://www.ucsc.edu/news_events/review/text_only/Winter-94/Something.html (15): http://www.strw.leidenuniv.nl/annrep02/node28.html (16): http://itss.raytheon.com/café/qadir/q1925.html (17): http://nedwww.ipac.caltech.edu/level5/Ashman2/Ashman3/html (18): http://vialattea.net/hubble/1994/9452_galform.htm (19):http://www.theblueplanet.ch/infocenter/articoli/febbraio2003/so16_buco_nero_230220 03.htm (20): http://www.cosediscienza.it/astro/05.%20LA%MATERIA%20OSCURA.htm IMAGE CREDIT Figure 1: Knut Lundmark (Lund Observatory), http://antwrp.gsfc.nasa.gov/apod/ap980523.html Figure 2,3,4,5,7: Cecchini G., IL CIELO, Utet, 1969 Figure 6: Anglo Australian Observatory, http://www.aao.gov.au/images.html Figure 8: NASA, http://adc.gsfc.nasa.gov/mw/mmw_images.html http://adc.gsfc.nasa.gov/mw/milkyway.html Figure 9: Russeil D., Observatoire de Marseille Figure 10: http://blueox.uoregon.edu/~courses/BrauImages/Chap23/FG23_019.jpg Figure 11: Hubble Space Telescope image of LMC-5 Figure 12: http://skyserver.sdss.org/dr1/en/sdss/discoveries/discoveries.asp Figure 13: http://www.anzwers.org/free/universe/sattelit.html Figure 14: http://antwrp.gsfc.nasa.gov/apod/ap970329.html A. Oksanen, 2.6 meter Nordic Optical Telescope Figure 15: http://www.anzwers.org/free/universe/galaxy.html Figure 16: http://zebu.uoregon.edu/~imamura/123/lecture-2/spiral.html Figure 17: http://www.space.com/php/multimedia/imagedisplay

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