PHYSICS OF ASTROPHSYICS - Energy

Transcription

1 PHYSICS OF ASTOPHSYICS - Enegy ENEGY esult of a foce acting though a distance. units = eg = dyne c i.e., foce x distance = g c /sec Two types: kinetic - enegy due to otion potential - stoed enegy due to position E = F d d v d vdt dt = = = a d kinetic v = d v vdt = 1 dt v 0 Vaious Fos of Kinetic Enegy Vaious Fos of Kinetic Enegy 1) Tanslational as above v 3) Theal KE = 1 v ) otational KE = 1 I I = oent of inetia = 5 E = 3 nkt Hee n is the nube of paticles, T, the tepeatue in Kelvins (K = C + 73) and k is Boltzann s constant: k = 1.38 x eg/degee K The kinetic enegy of a typical single paticle in a theal gas is = P fo a otating sphee of adius and ass with P, the peiod of otation (KE / paticle) = 1 pat v ando = 3 kt This ando speed is appoxiately equal to the speed of sound. 1

2 hee is the distance to the cente of, (pesued spheical) e.g. ai in the oo 1/ v = 3 kt v = 3kT 30 H = 5x10 3 g k = 1.38 x eg/k T= 93 K take ass,, fo to infinity foce acting though a distance v = 4.9 x 10 4 c/s (oe accuately 3 should be eplaced by = 1.4 fo ai) The speed of sound in ai at this tepeatue is actually 3.43 x 10 4 c/s (115 ft/s; 768 ph) E gav PE = 0 Potential enegy can be changed into kinetic enegy and vice vesa. PE less negative hee h 1 G PE = infinitely fa G PE = ( hh1) going fo h to h 1 PE negative hee h =PE

3 Suppose two asses "infinitely" fa apat have Escape Velocity PE = 0 no initial velocity with espect to one anothe. Total enegy = PE + KE = - G + 1 v =0 G PE = infinitely fa Now elease the two asses and let the fall towads one anothe until, which we shall conside to be vey sall, stikes the suface of at its adius,. Total enegy is "conseved", so 0 = - G + 1 v How uch speed would have to be given (staight up) to go to infinity and have no exta enegy (speed) left ove when it got thee? Hence v esc = G 3

4 Coets and asteoids typically ipact the eath with a speed of 11. to 70 k/s. That s 5,000 to 156,000 iles pe hou. The eath s obital speed is 30 k/s and a asteoid could be obiting in the opposite diection. The oldest known fossils of bacteia date fo 3.8 billion yeas ago. Fo alost the fist billion yeas ipacts ay have ade the Eath uninhabitable. But these sae collisions ay have bought soe of the cheicals necessay to life. (oigin of oceans debated - pobably not coets) Conside the ipact of even a 1 k diaete ock with density 5 g c -3 at 50 k s -1. Assuing a spheical shape, the ass would be 4/3 3, o.6 x g. The enegy, v = (0.5)(.6 x10 g)(5 x10 c/s) 8 = 3.3 x10 eg ~50 equivalent to about 780,000 egatons of high explosive. 4

5 Bainge eteo Cate Aizona, 1.19 k; 49,000 yeas Aoounga Cate, Chad Afica (Sahaa Deset) 17 k in diaete; 00 illion yeas Classic siple eteoite ipact cate iaged fo space, evidence fo ultiple ipacts anicouagan Cate, Quebec Canada 100 k, 1 illion yeas note tail of space shuttle Colubia, Lake is 70 k in diaete. one of the lagest ipact cates peseved on the suface of the Eath. Outline is a lake. Glacies have eoded uch of the oute stuctue. 5

6 Chicxulub Yucatan, Peninsula, exico Gavitational Binding Enegy 170 k acoss age illion yeas. Buied unde seveal hunded etes of sedient blocking it fo easy view (this pictue esults fo local gavitational and agnetic field vaiations). The asteoid that poduced this ipact cate is believed to have had a diaete of 10 to 0 k. The ipact hit a egion ich in sulfu beaing ock. The sky ay have been dak as night fo close to a yea. Tepeatues would have been feezing. Half the species on eath peished. Take this shell off What is the potential enegy of a spheical shell of ass sitting atop a sphee of adius and adius? That is how uch enegy would it take to eove the oute shell and take it to infinity? G E = << and what is the ass of that shell if the density is constant at all adii and the shell is vey thin with d <<. d +d G d de = = G d d=(4 )(d) = ( ) Let tot be the total ass of the sphee and its adius d Gd de = To eove a shell: 3 G( (4/3) )( 4 d) de = 3 = de = ( G)(4/3)(4)( )( ) d 0 = (16 / 3) 4 G d = (16 / 3) G( / 5) = (16 /15) G 3 3 but tot = (4 / 3) (4 / 3) 6 = (16 / 9) so 3 G = 5 tot o =

7 GAVITATIONAL BINDING ENEGY Defined as the total potential enegy of a gavitationally bound syste (note thee ae siila concepts based on the electic and stong foces - e.g., nuclea binding enegy Fo the sun = 3 5 = 0.6 G ( )( ) ( ) ( ) = eg (actually 6.9 x eg) The Kelvin-Helholtz tie scale (Lod Kelvin and Hean van Helholtz, id 1800 s) KH L 3G 5 L = 0.3 G = sec = 9.6 illion yeas (in fact, because the density is not constant, 0 to 30 illion yeas is close to coect) Lage ass stas have shote Kelvin-Helholtz tie scales. How fa could this go? Suppose contact to a black hole BH G = G c S G c Thee ae easons why this doesn t happen in odinay stas. In fact, this extee liit is neve achieved, but it is possible in soe cicustances to get 30% c. Indeed the gavitational binding enegy of a neuton sta is about 1/3 c and atte falling on neuton stas eleases about this uch enegy. Soe young stas, especially T-Taui stas, ae thought to get ost of thei cuent luinosity fo gavitational contaction, not nuclea fusion. 7

8 The Viial Theoe: Fo a syste bound togethe by a foce that is popotional to 1/, e.g., gavity, the total potential enegy is, in agnitude, equal to twice the total kinetic enegy (in all fos - heat, otion, and otation) KE = PE Always valid if the coponents of a gavitationally bound syste have been togethe a long tie, and ae not oving close to the speed of light o ae so dense as to be degeneate Assue: 1) A 1/ foce Outline of Poof ) The tie aveages of the kinetic and potential enegy ae well defined 3) The positions and velocities of all paticles ae bounded fo all tie * G = p i i p i = i vi * i dg dp = i dt i dt + i i vi d essentially G is the i total net angula oentu i dt and we assue that ove long intevals of tie but, d p i = F i and d i dt dt = v it is not changing i so dg = F i i + i v i since v i v i = v i dt i i So, F ight be e.g. dg = F i i + KE G = G dt i Now conside the tie aveage of both sides ove long peiods of tie Exaples: Obital otion G = v Planets aound the sun Stas bound to ilky Way Stas in a Globula Cluste v = KE = G = PE Whee did the othe half of the enegy go? dg dt 0 = PE + KE Theal kinetic enegy of a gavitationally bound gas 8

9 All of the gavitational enegy eleased as a sta - its total gavitational binding enegy has to go soewhee. Accoding to the Viial Theoe, half of the binding enegy gets adiated away as light. the othe half stays behind as heat. Thus appoxiately, PE = 3G 5 N 3 * kt = KE whee N * is the nube of atos in the sta, N* / ato 1 3 KE v kt paticle = paticle = The ass of a hydogen ato is 1/N A gas whee N A = 6.0 x 10 3 g -1 So G 5 N A kt T G 5kN A Note that as gets salle, T gets lage. (4.6 x 10 6 K fo the sun, which is not a bad estiate fo the aveage tepeatue. The cental tepeatue is about thee ties geate.) In fact this equation undeestiates T because the density of the sun is not constant. Note the iplications. Fo sta with constant ass,, contaction occus until T is high enough to bun a given fuel by nuclea eactions. When that fuel is gone, the sta - o pat of it - contacts futhe and the tepeatue goes up again. T G 5kN A Since = = 3 4 and T /3 1/3 1/3 (constant density assued to ake a siple aguent) T G log T(K) 7 6 highe T ignite nuclea fusion Log T 1/3 1/3 T /3 1/3 1/3 1/3 contaction 3 1 ideal P deg = P ideal degeneate T 1/3 fo a given T at a given is highe fo bigge lightest sta will be ass that hits this point. 9

10 Spial stuctue - siulation Hubble colo photogaphs: Pecession peiod 8.85 y fo obit Nodes point towads the sun evey 5.4 onths e = A node is whee the oon s obit cosses the plane of the ecliptic 10