AST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1
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1 Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be familia with. Section D is not pat of the couse and will not be examined - it is povided only fo you inteest. I have included Section D in the handout because it epesents one of the most impotant beakthoughs in science. Newton's demonstation that all of Keple's thee Laws of Planetay Motion could be explained by a simple foce f / and that this foce was the same phenomenon as familia teestial gavity was impotant because: (a) It was the fist time that the undelying cause of divese complex natual phenomena wee explained in tems of a single vey simple pocess, i.e. invese squae law gavity. (b) Gavity was the fist of the fundamental foces of Natue to be identified and descibed (even if we now know that Newtonian Gavity is only a vey good appoximation to gavity as descibed by Einstein's Geneal Relativity). (c) It was the fist time that pocesses in the emote Univese wee undestood in tems of the same phenomena seen on Eath. Newton's Gavity was a Univesal Foce. Thus the science of astophysics was bon. The fact that the Univese follows the same physical laws as ae seen in teestial laboatoies makes cosmology possible. In this handout, Sections A and B will be familia fom the lectues. Eveyone should be able to cope with the mathematics in Section C in which we deive the foce equied fo a body to move in a cicula path. The poof, in Section D, that Newtonian Gavity poduces obits which satisfy all of Keple's Thee Laws of Planetay motion, is enomously easie if we use vectos and vecto calculus. Many of you will not be familia with these tools. As stated above, I do not expect students to be able to epoduce Section D in an exam o elsewhee and you should ead it fo inteest only. I have povided a shot intoduction to vectos and vecto calculus fo those who will want to wok though this deivation. I hope many of you will ty this, and "expeience the beakthough" fist-hand. In Section E, I state without poof a couple of simple esults that Newton poved that enabled him to calculate the gavitational field nea to extended objects like the Eath. What you need to know fom this handout: (a) You should know Sections A and B, be able to apply the esults of Section E and be able to epoduce Section C. (b) In Section D, you need to know only (i) that the angula momentum, m d /, is conseved aound an obit. (ii) that the total enegy, E, is also conseved aound an obit d E mv m + m In the fist assignment, we will use these two esults, (i) and (ii), and we will obtain the same enegy equation (ii) when we come, late in the couse, to constuct the equation to descibe the evolution of an expanding Univese.
2 AST S: The oigin and evolution of the Univese Mathematical Handout : Obits in a simple gavitational field. A. Keple's Thee Laws of planetay motion Fom caeful analysis of Tycho Bahe's voluminous obsevations of the appaent positions of the planets as seen fom Eath, Keple fomulated his thee "laws" of planetay motion that descibed how the planets, including the Eath, moved in space. Keple's st Law: Each planets moves on an obit aound the Sun that is an ellipse with the Sun located at one focus of the ellipse Keple's nd Law: A line dawn fom the Sun to a given planet sweeps out equal aeas in equal time intevals as the planet moves aound the obit. Keple's 3d Law: Fo the vaious planets in the Sola System, thee is a elation between the peiod of the obit T (the time taken to complete one obit) and the semi-majo axis of the ellipse, 3 A: T A It should be noted that Keple did not undestand why the planets moved in this way though he suspected that it was elated to some teestial phenomenon and in fact was the fist to coin the tem astophysics. B. Newton's thee laws of motion Newton fomulated thee laws that descibed the motions of objects, based in pat on many of Galileo's ideas: Newton's st Law: Bodies move in a staight line at constant speed (i.e. at constant velocity) unless acted upon by a foce. Newton's nd Law: The magnitude of a foce is given by the mass and the acceleation the foce poduces. dv F m a whee a Newton's 3d Law: Fo evey foce thee is an equal and opposite eaction. C. Acceleation equied fo cicula motion The velocity of a paticle is continually changing as it moves aound a cicula path (emembe that velocity is a vecto quantity incopoating diectional infomation, as distinct fom speed, which does not) and it must theefoe, accoding to Newton's Fist and Second Laws, be acceleating and expeiencing a foce. Conside a paticle moving with speed v aound a cicula obit of adius. Let us look at the change in velocity while the paticle moves though a vey small angle aound the obit. Suppose that the paticle is initially moving in the y diection. In time t it moves a distance y coesponding to the angle.
3 cicula obit x motion in absence of foce y Now, in this same time inteval, the paticle must stop moving puely in y and must acquie a small velocity in the x diection, i.e. towads the cente of the obit (emembe we can make abitaily small). On the figue above, it moves x x y with a velocity x v x t t Theefoe the acceleation equied is a v t x t Now we can wite ( / t) as t d v whee w is the angula velocity of the obit (measued in adians pe second). Thus the acceleation equied to poduce a cicula obit is (.) v a Thus in ode to poduce a cicula obit we must continuously apply a foce, diected towads the cente of the obit, of m o mv /. A numbe of people, including Newton's geat ival Hooke as well as Newton himself, ealized that a foce field in the Sola System that fell off with the squae of the distance (and was popotional to the mass of the planet) would epoduce Keple's 3d Law. This is because the peiod, T, of a planet in a cicula obit is elated to by (.) T
4 Now, Keple's 3d Law states that 3 T since A fo a cicula obit Thus, fom (.) and (.) (.3) a Thus in the case of idealized cicula obits, the foce on a given planet must be popotional to the mass of the planet and invesely popotional to the squae of the distance between the Sun and the planet. But Keple's st Law states that planetay obits ae not cicles, but athe ae ellipses. Newton's geat achievement was to demonstate that such an invese squae foce field would also poduce elliptical obits, with the Sun at one focus, and with T A 3, theeby satisfying all of Keple's thee Laws descibing planetay motions. The cicula obits consideed above ae just a special case of moe geneal cicula obits. Futhemoe, he then showed that the foce that causes the Moon to obit the Eath was the same as causes apples to dop fom apple-tees. It is these two to which we now tun. D: Elliptical obits in a simple gavitational field (Section D is fo inteest only and is not a fomal pat of the couse) The deivation of the geneal obit in a invese-squae law gavitational field is most easily seen using vectos and vecto calculus: Box : Vectos and vecto calculus Fo those students who (a) ae not familia with vecto calculus and (b) wish to go though this next section, the following is a bief oveview of vectos and the calculus of them. Recall that wheeas many quantities, such as mass, speed, tempeatue, ae scalas (i.e. simply have a magnitude), othes ae vectos and have both magnitude and diection. Examples ae the displacement fom the oigin of some coodinate system, velocity, foce etc. We will wite vectos in bold pint and scalas (and the magnitudes of vectos) in egula pint. The diection of a vecto can be epesented by dividing it by its magnitude, i.e. a/a. The magnitude of a vecto can be epesented as a a Vectos can be combined in vaious ways. Addition (and subtaction) is staightfowad:
5 b a+b a Box (continued) Vectos can also be multiplied in two ways: The dot-poduct is a scala given by a. b ab cos whee is the angle between the vectos. The dot-poduct is the magnitude of one vecto times the component of the othe vecto that is paallel to the fist (and vice vesa). It is easy to see that a.b b.a. The coss-poduct is a vecto that has diection pependicula to both a and b and has magnitude a b ab sin The diection of a b is set by the ight-hand ule such that a, b and a b ae like the thumb, fist and middle finges of you ight hand. This means that ( a b) ( b a). Clealy paallel vectos (whee ) have zeo coss-poduct (as they must since the diection pependicula to both is then not defined!). Vecto diffeentiation occus in the nomal way. If we have a time dependent vecto a(t). Then da/. is given by da a( t + t) a( t) t a(t+)-a(t) a(t) a(t+) da/ is thus itself a vecto. Finally, diffeentiation of coss-poducts poceeds in the usual way by the chain ule d db da b ( a b) a +
6 Conside the simple case whee the gavitational field is dominated by a single point-like object of mass M (i.e. in the case of the Sola System, the Sun). Fo an obiting object of mass m, whose instantaneous position, elative to the cental object is given by and whose instantaneous velocity is v d/, Newtonian gavity descibes the foce and esulting acceleation of the obiting object as: (.4) m d m d v 3 If we take the coss poduct of this equation (.4) with, and cancel out m, we get: (.5) d GM since 3 If we now look at the quantity ( v) (i.e. ( d ) ) we get by definition d d d d d (.6) + fom (.5) and a a Thus the quantity ( v) is constant fo motion in a gavitational field. The quantity ( v) is the specific angula momentum (i.e. angula momentum pe unit mass), and the angula momentum,, is v. simply m ( ) (.7) ( ) constant m v Remembeing that the diection of ( v) is pependicula to and v, the constancy of ( v) that the obit lies in a plane. means Equation (.6) now explains Keple's Second Law since the ate at which aea swept out by the vecto is just da v (.8) sin ( v) If we now take the dot-poduct of equation (.4) with v ( d/), we get: (.9) m d v. v v. 3 With a few moments thought you should be able to convince youself that (.) d mv dv d d mv. and. 3 Theefoe, we can ewite (.9) as
7 d (.) mv You may ecognize that the quantity in backets is the total enegy, E, of the obiting object (i.e. kinetic enegy plus negative potential enegy). E mv Equation (.) theefoe simply poves the consevation of enegy. Because the obit is in a plane, fom (.6), we can conveniently adopt (, ) coodinates and dop the vecto notation. We have defined the angula momentum,, which we have seen is constant, to be (.) m v m d We can decompose v into a component paallel and pependicula to using Pythagoas. d d v + + d m Thus, we can ewite ou enegy equation as d (.3) E m + constant m Reaanging (.3) we get: (.4) d E m GM + m / Note in passing that the left hand side of (.4) is just the adial velocity of the paticle towads o away fom the cental object. If E is positive then this is still finite as the distance tends to infinity and the paticle can completely escape fom the influence of the cental object. On the othe hand, if E is negative, then the adial velocity must become zeo at some finite distance fom the cental object and in this case the paticle neve gets futhe away, and is bound to the cental object. The enegy E is thus often efeed to as the binding enegy of the paticle. Dividing (.4) by ( m d /) we get: (.5) m d d E m + GM m /
8 If we now make the following substitutions: u m du w u w then (.5) becomes dw d E m + (.6) ( w ) / w Equation (.6) has solution w w cos m i.e. substituting back into the above, (.7) ( + cos ) with d dw du constant (.8) E and + 3 G M m / The quantity is known as the eccenticity of the obit. A cicle is an ellipse with. Fo E <, i.e. fo an object which has negative net enegy (i.e. which is bound to the cental object), equation (.7) is in fact the equation of an ellipse with the oigin at one focus. Thus is Keple's Fist Law explained. You may be moe familia with the equation of an ellipse in Catesian coodinates: (.9) x a y + b In this case, the oigin of the Catesian system (x,y) is offset fom the oigin of the pola coodinate system (, ) by ( a,). The semi-majo and semi-mino axes of the ellipse ae given by:
9 (.9) a b ( ) / The E case coesponds to a paabola and the unbound E > case to a hypebola. The cicula case has E -G M m 3 / We can easily now ecove Keple's Thid Law by noting that the aea of an ellipse is (.) A ab Now, the time taken to go aound the obit is, fom (.8) (.) m ma T da m ab Now, fom (.8), we know /, and fom (.9) we have that b a, it follows that: (.) T 4 GM a 3 This is exactly Keple's Thid Law with a A. Note that the mass of the planet does not ente in to equation (.) so it applies fo any planetay system in which the gavitational field is dominated by a single massive object. E: Gavitational fields fom symmetically distibuted mass systems. Newton's analysis in Section D beautifully explained Keple's Thee Laws of planetay motion in tems of a simple / foce field emanating fom the Sun. Newton next wanted to demonstate that this was the same foce as familia gavity on Eath. Galileo's obsevations that the motions of objects on Eath was independent of thei masses, could be explained by Newton's new foce since the size of the foce was popotional to the mass of the body it was acting on, so the acceleation would be independent of mass. Howeve, fo seveal yeas he was stymied by the fact that the Eath is obviously not a point mass and he did not know how to calculate the gavitational field of an extended distibution of mass such as the Eath. Afte seveal yeas he was able to pove the following two theoems which will hee be stated without poof: Fo any spheically symmetic shell of matte, the gavitational field aising fom the mass on the shell is (a) zeo within the shell and (b) the same, outside the shell, as if all the mass of the shell was a point mass at the cente of the shell. Thus, the gavitational field at the suface of the Eath is the same as if the Eath's mass was concentated into a point at the cente of the Eath, so that one can in fact use the simple equation fo the point mass (.4). Futhemoe, the gavitational field at some adius within an extended spheically symmetic object (such as an idealized galaxy) is given by (.4) with M the mass enclosed within the adius. The effect of mass futhe out in the galaxy is zeo.
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