The Spiral Structure of NGC 3198.

Transcription

1 The Spial Stuctue of NGC Buce Rout Novembe 8, 2009 Abstact Obsevations of NGC 3198 show a discepancy between the otational velocity and its appaent geomety which defies the pedicted behaviou of Kepleian Dynamics. This pape econciles this anomaly by consideing the elativistic effect of gavity on galactic spial ams ove geat distances in a otating efeence fame. Kepleian dynamics hold tue in this analysis by consideing the otational behaviou of a cloud of stas as moe accuate than that of a cental mass with satellites at discete obits. A e-examination fom fist pinciples descibes the spial ams of NGC 3198 as a linea sta cloud of nea-unifom density which appeas, fom ou local efeence fame, as a non-unifom disc due to its otation. The appaent non-unifom adial distibution of stas is descibed by delayed gavitational inteactions ove geat distances in an acceleating efeence fame wheeby a unifom distibution of stas appeas to occupy an inceasing cicumfeence. The theoy is substantiated by deiving the shape of a linea sta cloud of the dimensions and otational velocity of NGC 3198 as it would appea fom Eath, using Einstein s equations and Kepleian dynamics. Since the deived shape is conguent with the obseved shape of NGC 3198, the exact shape and size of the esulting spial can be used to detemine its distance fom Eath with geat accuacy using simple tigonomety. Begeman[2] has povided an accuate and extensive measuement of the otational velocities of stas in the spial ams of NGC Unde the heading Discussion, Begeman states the following: A futhe analysis of the mass distibution in NGC 3198 has been given by van Albada et al (1985)[1]. The main conclusions fom that pape ae (i) the otation cuve of NGC 3198 can be descibed by a two-component mass model consisting of an exponential disk and a spheical halo, and (ii) the amount 1

2 of dak matte needed to explain the obseved otation cuve out to the last measued point is at least a facto 4 lage than the amount of visible matte. Accoding to fundamentals of otational dynamics and Newton s theoies of kinematics[6], the unbalanced foce acting on any body in cicula obit is diected constantly towads the cente of the body s obit. We feel it is easonable to assume cicula obits fo the stas in the spial ams. We conside an equivalent mass function, M(). The popeties of M() ae that it detemines an equivalent souce of gavitational attaction at the cente of the galaxy. This is a technique commonly used to simplify the mathematical calculations in gavitational poblems involving complex distibutions of matte known as finding the cente of mass of the matte distibution. Finding the cente of mass may be somewhat complex, nevetheless, an equivalent cente of mass function, M(), must exist because stas in the galactic ams obit aound it. We shall now detemine this distibution function. Conside the otational dynamics fomulae of a sta of mass m obiting an equivalent cente of mass M() and having a otational velocity v: v 2 m v2 m v2 m v2 v 2 = a = ma = F = G M()m 2 = G M() 2 v 2 = G M(). We can then wite: v 2 G = M(). (1) Theefoe, since v is constant thoughout the galactic ams, as detemined by the data, we must have: M() = κ (2) whee κ is a constant equal to v2. In ode to attempt to undestand the G convolutions and complexities of the M() mass distibution function of NGC 3198, we gaph it as in figue 1. Because of the geat distances involved, the time it takes fo the gavitational influence of the inne egion of this mass distibution function to each 2

3 M() Figue 1: Plot of M() vs. oute stas must be consideed. Consideing that changes in gavitational influence tavel at the speed of light, the obital displacement of the distibution function is popotional to the adial distance fom the cente of the galaxy. We have mathematically modeled this phenomenon and show it in figue 2. All matte aanged in this linea distibution, obiting about the cente of mass, will have the same obital velocity. This object appeas as a spial because it is vey lage and the speed of light is finite. This is pecisely what we see in the data. Any pesence of exotic mateial, such as dak matte, will invalidate the given data. Note the geomety of the matte distibution of the galaxy detemines the otational velocity pofile. A spheical matte distibution will esult in a otational velocity pofile popotional to 2 while a disk will esult in a otational velocity pofile popotional to. Since the otational velocity pofile is constant, the only matte distibution possible is one which is popotional to. This can only be the esult of a linea geomety. Futhemoe, it is a well established fact, and one I have obseved myself on many occasions, that ou own galaxy does not have spial ams as we look diectly into the cente of ou galaxy fom a vantage point in the southen hemisphee. All the stas of the spial am appea to be in a line of sight fom Eath into the cente. It seems ou spial am is staight while the ams of othe galaxies ae cuved. In essence, we ae looking into the nucleus of ou galaxy staight down a geodesic. In ou local efeence fame, the inne stas pull us towads the cente of the galaxy because they all appea lined up that way. Howeve, wee we to tavel millions of light yeas above ou galaxy, it 3

4 Figue 2: A spial geneated fom the effects of the geneal theoy of elativity on a galaxy having a adius of 46 kpc and whose stas all have an obital velocity of 151 Km s 1. The scale is in thousands of light yeas. Figue 3: NGC 3198 in Usa Majo. NGC 3198 is classified SBc(R).[9] would look like a spial just like vey many othe galaxies. The inne stas have moved along in thei obit by the time thei light and gavitational influence have eached us. See figue 7. Although the data invalidates the pesence of any mass distibution function othe than that given by figue 1, we do accept the validity of Begeman s measuements of otational velocity of the component stas within NGC 3198 as well as Begeman s eo analysis. This data is invaluable in obtaining an accuate measue of the distance to NGC A spial, in pola coodinates, is a emakably simple function. It is: (θ) = kθ. (3) Any two spials which can be laid one on top of the othe so that they coincide ae said to be conguent. Conguent figues ae said to be equal in all espects. If the paametes of two conguent spials ae equal, then the scales of the ectilinea sizes of the two spials ae also equal. The only paamete of any spial is the constant k in equation 3. We show that k depends only on the otational velocity of the galaxy to detemine its absolute size. We can then match its spial mophology, such as that displayed by a photogaph, to an independently geneated spial detemined by NGC 3198 s stella otational velocity and the geneal theoy of elativity. The essence and foundation of the geneal theoy of elativity is simply that gavitational influences tavel at 4

5 the speed of light[4]. This is all we equie of the theoy fo this measuement. Note that the photogaph has a ectilinea scale of minutes of ac while the geneated spial has a ectilinea scale in thousands of light yeas. Matching the two spials by scaling them so that they coincide allows us to detemine the distance to the galaxy. We have detemined that the galaxy subtends 7.33 ±.1 minutes of ac fo evey 136 ±4 kpc and have theefoe obtained the following data: 1. Distance = 19.6 ± 1.2 Mpc. 2. Radius = 180,000 ly. (55 kpc.) ±10% This is done as follows: we fist manipulate the position data of stas within NGC 3198 using a geometic pojection algoithm to obtain figue 4. To do this we otate an image of the galaxy 46 so that the majo axis of the galaxy is hoizontal and will not be alteed by futhe manipulations of the image. The mino axis is measued and compaed to the length of the majo axis. By inspection and using a spial ovelay oiented with the otated pictue of the galaxy, we measue that a vetical stetching facto of 2.65 will esult in a geometic tanslation equivalent to seeing the galaxy fom above. The image is a time lapse photo of the galaxy with a high degee of esolution. Even afte applying the equied vetical stetch to the image, individual stas can still be seen as a backgound dusting of stas along the lines of the visible spial ams of the galaxy. Also, thee is vey little pixilation. We obtain a pintout of the esultant figue 4 and set it aside. We then tun to geneating a spial ovelay. To constuct the spial we note the otational velocity of the points of the spial is a constant. Using equation 3 we need to find the paamete k. Conside an obseve at the cente of the galaxy obseving a sta at one light yea adial distance. Then = one light yea. Howeve, the sta has moved fo one yea in its obit and has tavesed v/c light yeas of ac length. Since we ae using adians, we have θ = (v/c). Futhemoe, ou scaling paamete to daw the spial in pola coodinates equies θ to be scaled 2π pe complete evolution and to be scaled as one light yea pe evolution fo a seies of concentic cicles making the scale of the spial to be dawn. We theefoe have: (θ) = 2πθ ( v (4) c) To have thousands of light yeas fo the adial scale, we have used: (θ) = 2πθ 1000 ( ) (5) v c 5

6 Figue 4: NGC 3198 with the positions of stas adjusted though a geometic pojection algoithm in ode to view the galaxy fom a efeence point diectly above the galaxy. Figue 5: A spial ule with size scale included. The axes ae in thousands of light yeas. We daw a spial, as in figue 5, which esults in a coodinate axis with a scale of thousands of light yeas using Maple, which we shall efe to as a spial ule. We then size figue 5, including coodinate axis, to match the size of the photogaph, and pint it out on a sheet of clea plastic. Placing the spial ule on top of the pintout of figue 4 and otating it about the mutual centes of the two pintouts, we achieve a emakably good match. See figue 6. Since the otational velocity of the galaxy as measued fom a local efeence point has been used to ceate both spials, that is: one fom the geneal theoy of elativity to detemine the shape of NGC 3198 fom the otational velocity of its component stas and the othe fom the constaints of the geneal theoy of elativity on the galaxy itself; they match. We theeby detemine the atio of the angula sepaation given by figue 4 to absolute distance as detemined by the coodinate axis of the spial ule and the est is staightfowad. Finding the distance to the galaxy is demonstated. We have detemined the atio of ly to 7.33 of ac. This is 41.7 kpc pe Then: R = π.122 whee R is the distance to the galaxy, o 19.6 Mpc. We estimate each of the ams to be about 180, 000 ly long. Included is figue 6 to demonstate the technique moe clealy. (6) 6

7 Conclusions We conclude: 1. NGC 3198 is a spial galaxy by inspection. Thee is no ba pesent in the galaxy. 2. NGC 3198 is gavitationally self bound in a linea mass distibution function of constant linea density. 3. NGC 3198 obits about its cente of mass in a single plane of obit. 4. The stas of NGC 3198 all obit with a constant otational velocity independent of thei espective distance fom the cente of otation. 5. Fom ovewhelming evidence in the obsevations of ou own galaxy, the linea mass distibution of NGC 3198 consists of stas and intestella dust and gas. Thee is no equiement fo exotic mateial in the linea mass distibution. Any mateial, exotic o othewise, having a geomety othe than that given by a linea mass distibution would invalidate a constant otational velocity pofile of mateial making up the galaxy. We theefoe conclude thee is no dak matte in NGC Although in the efeence fame of the stas of NGC 3198, the galaxy appeas as a linea mass distibution function, in the efeence fame of the Eath, the galaxy appeas as a spial. 7. NGC 3198 is 19.6 ± 1.2 Mpc. away fom us. 8. Einstein s geneal theoy of elativity and Newton s pinciples of gavitational attaction hold ove vey geat distances. 7

8 7.33 Figue 6: Ovelay of figue 4 with figue 5. The angula sepaation acoss the elatively bighte ends of the spial galaxy is indicated. The scale on the axis is in thousands of light yeas. 8

9 Figue 7: The cental egion of ou galaxy. Thee is no spial am stuctue visible since we ae looking down a geodesic into the cente. The geodesic is a line of sight staight down into the photogaph like looking into a deep well. The cental egion of the galaxy hee appeas at the bottom of the well. By the time the light and gavitational influence of the stas in the cental egion have eached us they have all moved in thei espective obits. In ou efeence fame, thee appeas to be a ball of stas suounding the cental egion. This ball of stas is pulling us towads the cente of the galaxy while ou tangential otational velocity keeps us in a cicula obit.[8] 9

10 Refeences [1] van Albada, T. S., Bahcall, J. N., Begeman, K., Sancisi, R., (1985), Astophys. J., #295, pp 305. [2] Begeman, K. G. (1989). HI Rotation Cuves of Spial Galaxies. Astonomy and Astophysics, #223, pp Retieved fom the High Enegy Astophysics Division at the Havad-Smithsonian Cente fo Astophysics. [3] Banad, S. and Child, J. M., Elements of Geomety, Pats I - VI, MacMillan & Co. Ltd., St. Matins Pess, New Yok, Fist edition, [4] Einstein, Albet, Lotntz, H. A., Weyl, H.. Minkowski, H., The Pinciple of Relativity Dove Publications Inc., New Yok, N.Y. (Fist published in 1923). [5] Einstein, Albet, The Foundation of the Geneal Theoy of Relativity, (pp in tanslation volume), Pinceton Univesity Pess, 1997, epinted fom The Collected Papes of Albet Einstein, Volume 6, The Belin Yeas: Witings, , A. J. Kox, Matin J. Klein, and Robet Schulmann, editos. [6] Newton, Si Isaac, E.A., Philosophi Natualis Pincipia Mathematica, [7] Chales W. Misne, Kip S. Thone, John Achibald Wheele, John Wheele, Kip Thone, Gavitation, W. H. Feeman; 2nd Pinting edition (Septembe 15, 1973). [8] Photo cedit: J. R. Henley, Stafield Obsevatoy, Nambou, Sunshine Coast, Queensland, Austalia, by pemission. [9] Photo cedit: Glen Youman, Penyn, Califonia, by pemission. 10