PHYSICS OF ASTOPHSYICS - Enegy http://apod.nasa.gov/apod/ ENEGY esult of a foce actng though a dstance. unts = eg = dyne c.e., foce x dstance = g c 2 /sec 2 Two types: knetc - enegy due to oton potental - stoed enegy due to poston E = F d d v d vdt dt = = = a d knetc v = d v vdt = 1 dt 2 v2 0 Vaous Fos of Knetc Enegy Vaous Fos of Knetc Enegy 1) Tanslatonal as above v 3) Theal KE = 1 2 v2 2) otatonal KE = 1 2 I 2 I = oent of neta = 2 5 2 E = 3 2 nkt Hee n s the nube of patcles, T, the tepeatue n Kelvns (K = C + 273) and k s Boltzanns constant: k = 1.38 x 10-16 eg/degee K The knetc enegy of a typcal sngle patcle n a theal gas s = 2 P fo a otatng sphee of adus and ass wth P, the peod of otaton (KE / patcle) = 1 2 pat 2 v ando = 3 2 kt Ths ando speed s appoxately equal to the speed of sound.
hee s the dstance to the cente of, (pesued sphecal) e.g. a n the oo 1/2 v 2 = 3 2 kt v = 3kT 30 H = 5x10 23 g k = 1.38 x 10-16 T= 293 K eg/k take ass,, fo to nfnty foce actng though a dstance v = 4.9 x 10 4 c/s (oe accuately 3 should be eplaced by = 1.4 fo a) E gav The speed of sound n a at ths tepeatue s actually 3.43 x 10 4 c/s (1125 ft/s; 768 ph) PE = 0 Potental enegy can be changed nto knetc enegy and vce vesa. PE less negatve hee h 1 PE = G nfntely fa G PE = 2 ( h2h1) gong fo h 2 to h 1 PE negatve hee h 2 =PE
Suppose two asses "nfntely" fa apat have Escape Velocty PE = 0 no ntal velocty wth espect to one anothe. Total enegy = PE + KE = - G + 1 2 v2 =0 G PE = nfntely fa Now elease the two asses and let the fall towads one anothe untl, whch we shall consde to be vey sall, stkes the suface of at ts adus,. Total enegy s "conseved", so 0 = - G + 1 2 v2 How uch speed would have to be gven (staght up) to go to nfnty and have no exta enegy (speed) left ove when t got thee? Hence v esc = 2G
Coets and asteods typcally pact the eath wth a speed of 11.2 to 70 k/s. Thats 25,000 to 156,000 les pe hou. The eaths obtal speed s 30 k/s and a asteod could be obtng n the opposte decton. Hstocal Ipacts The oldest known fossls of bactea date fo 3.8 bllon yeas ago. Fo alost the fst bllon yeas pacts ay have ade the Eath unnhabtable. But these sae collsons ay have bought soe of the checals necessay to lfe. (ogn of oceans debated - pobably not coets) Consde the pact of even a 1 k daete ock wth densty 5 g c -3 at 50 k s -1. Assung a sphecal shape, the ass would be 4/3 3, o 2.6 x 10 15 g. The enegy, 1 2 v2 =(0.5)(2.6 x10 15 g)(5x10 6 c/s) 2 nb. =510 4 c =3.3x10 28 eg ~50 equvalent to about 780,000 egatons of hgh explosve.
Bange eteo Cate Azona, 1.19 k cate; 49,000 yeas Ogn debated fo decades http://en.wkpeda.og/wk/eteo_cate Aoounga Cate, Chad Afca (Sahaa Deset) 17 k n daete; 200 llon yeas Classc sple eteote pact cate ~50 pacto N-Fe about 10 egatons eteote ostly vapozed aged fo space, evdence fo ultple pacts ancouagan Cate, Quebec Canada 100 k, 212 llon yeas note tal of space shuttle Coluba, 1983. Lake s 70 k n daete. one of the lagest pact cates peseved on the suface of the Eath. Outlne s a lake. Glaces have eoded uch of the oute stuctue.
Chcxulub Yucatan, Pennsula, exco Gavtatonal Bndng Enegy Cate 170 k acoss age 64.98 llon yeas. Bued unde seveal hunded etes of sedent blockng t fo easy vew (ths pctue esults fo local gavtatonal and agnetc feld vaatons). The asteod that poduced ths pact cate s beleved to have had a daete of 10 to 20 k. The pact ht a egon ch n sulfu beang ock. The sky ay have been dak as nght fo close to a yea. Tepeatues would have been feezng. Half the speces on eath peshed. Take ths shell off What s the potental enegy of a sphecal shell of ass sttng atop a sphee of adus and adus? That s how uch enegy would t take to eove the oute shell and take t to nfnty? G E = << and what s the ass of that shell f the densty s constant at all ad and the shell s vey thn wth d <<. d +d G ()d de = = G 4 3 3 4 2 d d=(4 2 )(d) () = 4 3 3 ( ) Let tot be the total ass of the sphee and ts adus d To eove a shell: 3 2 G( (4/3) )( 4 d) de = 3 2 2 2 = de = ( G)(4/3)(4)( )( ) d 0 = (16 / 3) Gd de = 2 2 4 G 0 d 2 5 2 5 2 = (16 / 3) G( / 5) = (16 /15) G 2 3 3 but tot = (4 / 3) (4 / 3) 2 2 6 = (16 / 9) so 2 3 G tot = 5 o 16 16 3 = 15 9 5
GAVITATIONAL BINDING ENEGY Defned as the total potental enegy of a gavtatonally bound syste (note thee ae sla concepts based on the electc and stong foces - e.g., nuclea bndng enegy Fo the sun = 3 5 = 0.6 2 G ( )( 1.989 10 33 ) 2 ( 6.96 10 10 ) ( ) 6.67 108 = 2.3 10 48 eg (actually 6.9 x 10 48 eg) The Kelvn-Helholtz te scale (Lod Kelvn and Hean van Helholtz, d 1800s) KH 2L 3G 2 = 0.3 G 2 5 2 L L = 3.0310 14 sec = 9.6 llon yeas (n fact, because the densty s not constant, 20 to 30 llon yeas s close to coect) Lage ass stas have shote Kelvn-Helholtz te scales because L nceases faste wth than 3. How fa could ths go? Suppose contact to a black hole BH G 2 = G 2 c 2 S 2G c2 Thee ae easons why ths doesnt happen n odnay stas. In fact, ths extee lt s neve acheved, but t s possble n soe ccustances to get 30% c 2. Indeed the gavtatonal bndng enegy of a neuton sta s about 1/3 c 2 and atte fallng on neuton stas eleases about ths uch enegy. As we shall see, t s ths enoous bndng enegy of neuton stas that powes supenovae. Soe young stas, especally T-Tau stas, ae thought to get ost of the cuent lunosty fo gavtatonal contacton, not nuclea fuson.
The Val Theoe: Fo a syste bound togethe by a foce that s popotonal to 1/ 2, e.g., gavty, the total potental enegy s, n agntude, equal to twce the total knetc enegy (n all fos - heat, oton, and otaton) 2 KE = PE Always vald f the coponents of a gavtatonally bound syste have been togethe a long te, and ae not ovng close to the speed of lght o ae so dense as to be degeneate Assue: 1) A 1/ 2 foce Outlne of Poof http://ath.uc.edu/hoe/baez/val.htl 2) The te aveages of the knetc and potental enegy ae well defned 3) The postons and veloctes of all patcles ae bounded fo all te * T = p p = v * dt dp = dt dt + v d essentally T s the total net angula oentu dt and we assue that ove long ntevals of te but, d p = F and d dt dt = v t s not changng so dt = F 2 + v snce v v 2 = v dt So, F ght be e.g. dt = F + 2KE G = G dt 2 Now consde the te aveage of both sdes ove long peods of te Exaples: Obtal oton G 2 = v2 Planets aound the sun v 2 = G Stas bound to lky Way Stas n a Globula Cluste = PE =2KE Whee dd the othe half of the enegy go? dt dt 0 = PE + 2 KE Theal knetc enegy of a gavtatonally bound gas
All of the gavtatonal enegy eleased as a sta - ts total gavtatonal bndng enegy has to go soewhee. Accodng to the Val Theoe, half of the bndng enegy gets adated away as lght. the othe half stays behnd as heat. Thus appoxately, PE = 3G 2 5 2N 3 * 2 kt = 2 KE whee N * s the nube of atos n the sta, N* / ato 1 3 KE v kt 2 patcle = patcle = 2 2 The ass of a hydogen ato s 1/N A gas whee N A = 6.02 x 10 23 g -1 so the nube of atos n the sta, N *, s oughly N A. So G 2 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, whch s not a bad estate fo the aveage tepeatue. The cental tepeatue s about thee tes geate.) In fact ths equaton undeestates T because the densty of the sun s not constant. Note the plcatons. Fo sta wth constant ass,, contacton occus untl T s hgh enough to bun a gven fuel by nuclea eactons. When that fuel s gone, the sta - o pat of t - contacts futhe and the tepeatue goes up agan. T G 5kN A Snce 4 3 3 3 4 and T 2/3 1/3 1/3 (constant densty assued to ake a sple aguent) T G log T(K) 7 6 hghe T gnte nuclea fuson -1 0 1 2 3 4 Log 4 3 3 T 1/3 1/3 T 2/3 1/3 1/3 1/3 3 2 1 P deg = P deal T 1/3 fo a gven T at a gven s hghe fo bgge lghtest sta wll be ass that hts ths pont.