15 ρ R Total angular momentum of the Jupiter-Sun System. momentum plus the spin momenta of Jupiter and the Sun.

Transcription

1 4-40 I I I I ρ x + y Ž d V I I I ρ x + y Ž ddω I I ρ sin θ Ž d sinθ Ž d θ π I I ρ 4 cos θ Ž d cos θ Ž d 8 π 3 I ρ 4 d 5 MR 8 π 5 ρ R 5 47 Total angula momentum of the Jupite-Sun System The total angula momentum is the sum of the obital (otational) angula momentum plus the spin momenta of Jupite and the Sun Jupite is the lagest planet in the sola system with mass M J M u and adius R J cm The semi-majo axis of Jupite s obit is a 5 AU, its obital eccenticity ε 0048 and its obital peiod is P 86 y Its obital angula momentum is given by (see 4-30) L µg{(m u + M J ) a (-ε ) )} / g cm s - The distance of the Sun fom the cente of mass is

2 4-4 a u am J M J + M u cm (slightly lage than the Sun s adius) Assuming that the Sun moves in a cicula obit, its angula momentum is L u M u v u a u M π a u u P (Note: L u is not the sola luminosity hee!) g cm s - The distance fom Jupite to the cente of mass is a J a - a u ~ a So its angula momentum is L J M J v J a J M J M π a u P cm s - So L L u + L J (as it must) The spin angula momentum of the Sun is L u s I u ω u I u π P u 4 π 5 M u R u P u P u 6 days, R u km, M u g

3 4-4 L u s g cm s - Similaly, using P J 0 hous L J s g cm s - Angula momentum of the Sun-Jupite system is dominated by the obital motion but if the Sun wee a apidly otating sta, its spin could dominate 43 Binay stas (See Ostlie and Caoll, Chapte 7) Moe than half the stas ae multiple stas, most of which ae binay pais Fo sola-type stas, the obseved atios of single: double: tiple: quadupole systems is 45:46:8: Thee ae seveal classes of binaies: visual binaies: both can be detected obiting in ellipses about one anothe (Siius is a famous example - Siius A is a main sequence sta of spectal type A, Siius B is a white dwaf of specta type A5 The peiod is 499 yeas Siius was fist discoveed as an astometic binay Astometic binaies ae binaies in which only one sta is obseved but its motion is oscillatoy, indicating the petubing pesence of a dim companion Spectoscopic binaies ae visually unesolved but peiodic oscillations occu in thei spectum If only one stella spectum is obseved, the binay is single-lined; if both ae obseved, the binay is double-lined

4 4-43 Eclipsing binaies occu when the two stas eclipse one anothe, poducing peiodic changes in appaent bightness Peiods of binay stas vay fom a few hous to hundeds of yeas Fom data on the peiods we can use the law of gavitation to infe masses Conside a double-lined spectoscopic binay The specta of the two stas ae supeimposed We can use Dopple shifts to measue the adial velocities of each sta, even though they may be too close fo thei obits to be distinguished Fig 4- The line joining the stas is otating with angula velocity Ω and K and

5 4-44 K ae the infeed adial velocities The figue shows the individual adial velocities Fom it we obtain the peak velocities of each sta and the binay peiod τ If the shape is accuately sinusoidal, the obits ae cicula with ε 0 The distances and fom the cente of mass ae constant fo cicula obits The cente of mass is the cente of the obits of both stas and of thei elative motion Ω M cm M Ω Fig 4-3 M M + M M M + M The distance of M fom M is + The peiod and the sepaation ae elated by Keple s Thid Law

6 4-45 τ 4 π 3 G M 4 π + Ž 3 G M + M Ž and the speeds of the stas ae v Ω, v Ω whee Ω is the angula velocity Ω π τ So + (v + v )/Ω M M v v M + M Ž Ω 3, independent of i The peak velocities equal v and v only if the obital plane is paallel to the line of sight If i is the inclination angle between the line of sight and the nomal to the obital plane the angle between the plane of the sky (defined as pependicula to the line of sight) and the plane of the obit G n v o i K v Fig 4-4

7 4-46 K v sin i Ω sin i π τ sin i K v sin i Ω sin i π τ sin i If i 0, we obseve zeo velocities (no infomation) i 90 is edge on K v, K v The mass atio is in any case independent of i M M K K v v The sepaation + τ K + K Ž π sin i The total mass we obtain fom M Ω 3 G 4 π 3 τ G M M + M τ π G K + K Ž 3 sin 3 i Hence we may wite fo the case when only K can be measued,

8 4-47 M + M τ π G sin 3 i M + M Ž 3 M 3 K 3 M + M τ π G sin 3 i Š Œ M + 3 K 3 M If we wite this equation in the fom τ π G K 3 M sin i Ž 3 M + M Ž f ( M ) we define a function f(m) called the mass function The left-hand side depends only on obsevable popeties and is useful fo statistical studies (Ostlie and Caoll p ) We may wite fo the sepaate masses, M sin 3 i τ π G K + K Ž K M sin 3 i but in geneal we do not know i τ π G K + K Ž K Fo eclipsing binaies, each sta successively eclipses the othe To see them, i must be nea 90, assuming that the stella adii ae much less than the stella sepaation Masses ae insensitive to i fo i nea 90 since sin i ~

9 4-48 t a t c t b c b a Fig 4-5 Fom the duation of the eclipses we can infe the adii of the stas Assume i 90 Let t a be the time of fist contact of the pimay eclipse and let t b be the time of minimum light Duing eclipses stas ae moving nealy pependicula to the line of sight to the obseve in opposite diections Thus, if v s and v l ae the velocities of the small and lage stas espectively, thei elative velocity is v s + v l The adius of the smalle sta is given by

10 4-49 s (v s + v l )(t b - t a ) and the adius of the lage sta by l (v s + v l )(t c - t b ) whee t c - t b is the duation of minimum light (see Ostlie and Caoll 7) 44 Extasola planets It is pimaily by velocity measuements of the paent sta that exta sola planets have been detected and we do not know the planet velocity Let M s be the stella mass and M p the planetay mass Suppose τ is the measued peiod and K s the measued stella velocity The mass function f (M) is M p sin i Ž 3 τ K 3 s M p + M s Ž π G If we assume M s >> M p, M p sin i Ž 3 τ K 3 s π G M s Obsevations of the sta Peg A a G sta like the Sun have evealed the existence of a companion and τ and K s have been measued Fom them we get

11 4-50 M p sin i ( M s ) /3 The mass of Peg A is 095 M u ~ 0 30 kg so if sin i M p kg It is customay to expess the mass of exta sola planets in Jupite masses M p 045 M J The obital adius is obtained fom a K s τ / π It equals 3000 km Fom a p a - M s M p, we obtain fo the distance of the planet fom the sta a value of 005 AU The value of M p and a p depend on the assumption that i 90 M p could be undeestimated and a p oveestimated 45 Supenovae in binay systems Supenovae ae exploding stas Befoe explosion many occu as binay systems and ae caused by mass flow fom a companion sta What happens to the binay system when the explosion occus and the mass of one sta is educed, possibly to zeo?

12 4-5 Befoe, thee ae two stas in cicula obit v M M CM v Fig 4-6 Assume cente of mass is at est, take it as oigin M + M 0 M v + M v 0 M > M explodes, leaving new mass M M - M Remaining binay is not at est In a spheical explosion, linea momentum caied away is zeo If v c is the new CM velocity, momentum consevation yields M v + M v (M + M )v cm (v and v ae the same immediately befoe and afte the explosion and total momentum is the same as the total mass moving with the cm velocity) Using v -M v /M and M M - M,

13 4-5 (M - M) Š Œ M v M + M v (M - M + M )v cm giving v cm M M M M + M M Ž v If M M, v c v (as it must) Typical values ae M 0 M u, M 5 M u, M 85 M u Then M 5 M u (appopiate fo a neuton sta) v cm 8 5 x 5 0 x 5 5 v 0 77 v Fo close binaies, v may be seveal hunded km s - so system eally moves To detemine whethe o not the binay emains bound, calculate the binding enegy, that is, the intenal enegy without the cente of mass enegy The total intenal enegy of the system immediately afte the explosion is

14 4-53 E N Š Œ M N v + M v M N + M Ž v N G M cm Total kinetic enegy Enegy of CM motion Gavitational Enegy M Now G ( M + M ) v v Ž fo cicula obits We obtain, witing eveything in tems of v E N M M Ž Š Œ M v + M M v M + M M Ž : M M v ; < M M + M M Ž B E C E D M M Ž M : ; M + M + M B E C < M E v D which (believe it o not!) simplifies E N M v Ž M M Ž M + M Ž M M + M M Ž M + M M Ž

15 4-54 All tems ae positive except the last facto Thus fo E to be positive (no binding), mass ejected M > / (M + M ) In the numeical example on p > (0 + 5) and the neuton sta depats at high velocity Pulsas (otating neuton stas) often have high velocity as they leave the galactic plane The esult can be obtained moe eadily using the CM system in which the total enegy is E tot M v cm + µ ( v -GM/) whee v is the elative velocity Befoe the explosion fo the initial cicula obit and v GM E µ v GM GM (see page 4-9) whee E is intenal enegy

16 4-55 Afte explosion, v is unchanged µ, M and E change as M changes to M Intenal enegy is changed fom v GM to E µ N Š Œ v GMN GM GMN So intenal enegy > 0 if M < M/ and ejected mass M > M/ 46 Tides When two bodies ae in obit aound each othe, the othewise spheically symmetic gavitational field is distoted by the gavitational attaction of the othe body The foce field can be chaacteized by equipotentials which ae like contous of height on a map; the foce is zeo tangent to the equipotential suface and is nomal eveywhee to the suface

17 Weak tides Eath Moon R Fig 4-7 Fo the Eath-Moon system, the Moon pulls the nea suface most stongly, the cente of the Eath less stongly and the fa suface least stongly The diffeential foce gives ise to ocean tides The ocean suface adjusts to become an equipotential The potential is fomed by the gavitational attaction and by the centifugal foce that aises because the Eath-Moon system is obiting about the cente of mass Assume masses ae concentated at the centes of the Eath and Moon The gavitational potential at a point fo a unit test mass is V ( ) GM GM to which must be added the centifugal potential aising because of the otating fame It is (fom 4-9)

18 4-57 θ ω / whee ω is the angula velocity ω G ( M + M ) 3 R R being the Eath-Moon distance (Remembe µ E µ θ + G µ M We fist evaluate V() at a point E µ ω + G µ M ) b Μ a θ CM D Μ Moon R Fig 4-7 Moon, Eath ae small compaed to Eath-Moon distance R so we wite

19 4-58 GM GM D GM R + a ar cos θ Ž / GM R Š Œ + a R a R cos θ / Binomial expansion + x Ž / x x + 0 x 3 Ž Thus GM D - GM R Š Œ a R + a R cos θ + 3 a cos θ R + O Š Œ a 3 R GM R : ; a + < R P cos θ Ž + a R B P cos θ E Ž + C D E (Altenatively use expansion R a R 3 n Š Œ a R n P n ( cos θ ) R > a whee P n (cos θ) ae Legende polynomials) The tem in the potential a R cos θ is linea in z, whee z is the diection fom M to M so its gadient descibes a constant foce GM /R which must be canceled by the centifugal potential The M centifugal potential can be witten, using R, M + M

20 4-59 ω ω a ω + a a cos θ Ž ω Š Œ M R M + M + a M M + M R a cos θ So adding this we have fo the total potential Φ() Φ ( ) GM a GM R + a R cos θ + / ( 3 cos θ ) a R G M + M Ž R 3 Š Œ M R + a Š Œ M R a cos θ M + M M + M The tem in a R cos θ is indeed canceled out by the centifugal foce that keeps the body in a cicula obit The gadient of tems that do not depend on a o θ is zeo, so they may be omitted and we have fo the local tidal potential Φ ( ) GM a Ga R 3 3 M cos θ + M Ž Expand a in tems of its height above the mean sea level a R + h Then

21 4-60 a R Š Œ + h, R a R Š Œ h R and Φ ( h, θ ) GM R Š Œ h R G R 3 R Š Œ + h R 3 M cos θ + M Ž - GM R h 3 Š Œ M Š Œ R 3 R M R cos θ ignoing constant tems GM R g acceleation due to gavity at the suface of the Eath The suface will adjust to be an equipotential (the tangential foce vanishes) so

22 4-6 Φ ( h, θ ) constant h 3 Š Œ M Š Œ R 3 R M R cos θ + constant The height of the tides is the diffeence between high and low values of h Since cos θ vaies between and 0, we get fo the height of the tides with M M 8, R 6000 km adius of Eath R 380, 000 km Eath Moon distance Then h 54 cm The same calculation with the Sun in place of the Moon yields h 3 x (33000) x Š Œ x 6400 km 5 cm 5 x 0 8 (pesumably by chance, they ae of the same ode) The tidal effects combine vectoially When the Moon is at conjunction o opposition, the two foces add to cause high tides

23 Tidal fiction The continents ae pulled though the ocean bulges and the tidal bulge is dagged ahead by the spinning Eath Thee is a loss of enegy by fiction and the spin of the Eath is slowed The day is getting longe (Thee is evidence fom gowth scales in fossil coals that thee wee 400 days in a yea about 00 million yeas ago) Angula momentum is conseved so the Moon inceases its angula momentum It can do so because the non-symmetic bulge ceates a gavitational toque back on the Moon Inceasing the angula momentum means the Moon must move outwad ( o L /Gmµ, a o ) and so the month is getting ( -ε ) longe The lowest enegy state of the Moon-Eath system is one in which the Eath and Moon pesent the same face in which case the tidal distotion will have eached its equilibium shape that involves no elative motion of any mateial The Eath and Moon will be tidally locked and thee will be no dag The tidal bulge will point diectly at the Moon and the Eath and the Moon will cootate Because the Moon is not exactly spheical, patial locking has aleady occued, in that the Moon otates with the Eath so that it shows the same face all the time The Moon is in synchonous otation such that the obital peiod of the Moon aound the Eath equals the otation o spin peiod of the Moon The ultimate equilibium caused by tidal fiction is that in which the spin velocity of the Eath equals the angula velocity of the Moon in obit aound the Eath (o the angula velocity of the Eath about the moon) so that month equals day

24 4-63 To calculate when that equilibium will be eached, use the consevation of angula momentum The angula momentum of the Eath-Moon system is the sum of the angula momentum of the spinning Eath and Moon and the angula momentum of the Moon s obit aound the Eath The angula momentum of the Eath may be witten I ω whee I is the moment of inetia and ω is the spin angula velocity Fom p 4-40, I MR 5 The total angula momentum is I ω + I ω + M M M ω R o (see 47) (note the Moon spin and obital angula velocities ae equal) and eventually is I ω f + I ω f + M M M ω f R f whee ω f π/(ultimate day o month), R f the ultimate Eath-Moon distance efes to the Eath and to the Moon and R o 380,000 km is the pesent Eath-Moon distance Keple s law gives R f R f R Š Œ ω ω f / 3

25 4-64 Now M ~ 83 M so M M M ~ M Also I ω is small compaed to I ω and as we can check once we have the answe, both spins ae negligible in the final state Then the initial angula momentum can be appoximated by 5 M R ω + M ω R o and the final angula momentum by Š M ω f R f M ω R Œ ω o ω f Hence / 3 ω ω f : ; < 5 M R ω + M ω R o M ω R o B E C E D 3 : ; < + 5 Š Œ M Š Œ R Š Œ ω B E C M R o ω E D 3 : ; < + 5 H 83 Š Œ 6400 E H 8 B C 380,000 E D 99 3

26 4-65 The final length of the day and month will be 8 x days The cuent lengthening of the day is about 0 days in 0 9 yeas so it will take moe than 0 0 yeas to each equilibium (We will have been engulfed by the Sun in its evolution by then) The same tidal foces bing binay stas into cootation, tidally locked to each othe 47 Roche stability limits fo satellites Objects can be ton apat by tidal foces We give an appoximate desciption Tidal potential at the point (,θ) is Φ (, θ ) GM G R 3 3 M cos θ + M Ž + constant Suppose M is the mass of a small satellite obiting a lage paent sta o planet of mass M M << M M Α z Fig 4-8

27 4-66 Calculate the foce at point A If it points towads the cente of the satellite, it is a estoing foce If it points away towads M, the satellite is ton apat Gavitational acceleation along the z axis at A is given by g z L z Φ d dz GM z Gz R 3 3 M Ž GM z + z 3 GM z 3 R 3 Gz Š Œ M + z 3 3 M R 3 The condition fo Roche stability is M > 3 M 3 R 3 Putting z, fo a satellite of mean density ρ we obtain ρ > 9 4 π M R 3 o no satellite of density ρ is stable inside the Roche adius R cit The citical adius is

28 4-67 R cit Š Œ 9 M / 3 4 π ρ If the paent (planet) and satellite have same density R cit 3 / 3 R 44 R whee R is adius of the paent (Roche calculated 44 R using a model of the selfgavity of the satellite) This is essentially the physics of the ings of Satun (and othe planets) Mateial within the Roche limit cannot fom bodies such as moons because of the disuptive effect of tidal foces 48 Roche lobes Conside a binay stella system in a cicula obit The intesections of the equipotential sufaces with the plane of the obit ae shown in the Figue (See Ostlie and Caoll p 687)

29 4-68 The Roche equipotential sufaces plotted in the equatoial plane fo two point mass with a mass atio equal to /3 The shot aows indicate the diection of the effective gavitational field in the fame of efeence which cootates with the obital motion The effective gavity vanishes at the five Lagangian points L, L, L 3, L 4, L 5 The fist thee, L, L, L 3, lie along the line joining the two mass points; the last two, L 4, L 5, fom equilateal tiangles with the two mass points, M and M The sideways figue 8 which passes though the L point contains the two Roche lobes Fig 4-9 Thee ae five stationay points, called Lagangian points, whee the foce vanishes Close to each sta, the equipotentials ae dominated by the gavitational attaction and the equipotentials ae cicles centeed at the stas (taken to be point souces) Fa fom the stas, the equipotentials ae dominated by the outwadly diected centifugal foce Thee the equipotentials intesect the equatoial plane in cicles enclosing both stas The two kinds of equipotentials ae sepaated by Roche

30 4-69 lobes aound each sta indicated by the figue of eight The Roche lobes intesect at a saddle point, foming in a pitche-like shape Roche lobes can be used to futhe classify close binaies If both stas ae smalle than thei Roche lobes, the system is a detached binay If one fills its Roche lobe, the system is a semi-detached binay and matte will flow though the contact point If both stas fill thei Roche lobes they ae contact binaies and they have a common envelope The Roche lobe is the maximum possible size of the sta If a sta of mass M becomes lage than its Roche lobe, it oveflows and dumps mass though the saddle-point on to the companion sta A common scenaio is the case whee M is initially much lage than M (possibly also losing mass to infinity in a stella wind) Mass flows fom M to M Eventually M becomes a white dwaf and cools M has gained mass and so it evolves faste and oveflows back on to M This pocess manifests itself in an X-ay souce As the white dwaf accumulates mass, it may be foced into a gavitational collapse to a neuton sta in a supenova explosion 48 Effect of mass tansfe on binay obits Suppose M is filling its Roche lobe and dumping mass on to M Mass and angula momentum ae conseved but not enegy The mass is heated and dissipates enegy in adiation M is gaining mass so M M > 0 The angula momentum fo a

31 4-70 cicula obit L µ R ω µ R Š Œ GM R 3 / M M R / G / M M / M R / Ž G / M / Ž 0 dl dt G / M / Ž Š Œ M M R / + M M R / + G / M / Ž M M R R / Solving fo R and eliminating M in favo of M, we obtain R R Š Œ M M M M M If the lighte sta M is losing mass R > 0 and stas daw apat Often this teminates the mass flow since it puts M deepe into its Roche lobe Altenatively the mass tansfe poceeds slowly on the stella evolutionay time scale that it takes M to fill its inceasingly lage Roche lobe If the heavie sta is losing mass R is negative The stas get neae which

32 4-7 inceases mass flow leading to a catastophic instability In pactice fiction leads to a mege of the two stas 49 The Viial Theoem Hee I pove a useful theoem, the viial theoem Intoduce I N 3 m i i i (simila to moment of inetia but about a point) The kinetic enegy of N inteacting paticles of masses m i and velocities v i is T N 3 m i v i i The gavitational potential enegy is V 3 3 j G m i m j i j Newton s law m i v i L i V

33 4-7 Diffeentiate I with espect to time twice I N 3 m i v i A i i I N 3 m i v i A i + i N 3 m i v i A v i i 3 i A L i V + T i This is the time-dependent viial theoem To evaluate 3 i A L i V i conside the scaled potential V(λ i ) obtained by eplacing all position vectos by λ i ρ i, say Then dρ i /dλ i and d d λ V λ i Ž d d λ V ρ i Ž d V ρ i Ž d ρ i Ž d ρ i d λ L i V A i Thus

34 4-73 d d λ V λ Ž 3 i i A L i V Fo a gavitational potential V λ Ž λ V Ž so 3 i A L i V i λ V Ž Put λ Then 3 i i A L i V V Ž and I V + T (See Ostlie and Caoll pp 53-56) If a gavitational system is in equilibium, neithe inceasing o deceasing in size, it must have the long time aveage values < V > and <T > such that < V + T> 0, < V > - <T > We can pove this by aveaging ove a long time Γ

35 4-74 V + T Γ Γ I 0 V + T ) dt Ž Γ I Γ Ž I 0 Ž If Γ is the obital peiod, I ( T ) I ( 0 ) Moe geneally, if all paticles emain bounded with bounded velocities fo all time, I(t) emains bounded and the ighthand side tends to zeo (This elationship <T> -< V> applies also to the kinetic and potential enegies of many electon atomic systems bound by the Coulomb attaction between the nucleus and the electons and can be established using quantum mechanics) (Fo a hamonic oscillato, V ~, V(λ) ~ λ V() d V(λ) λv() d λ 3 i V i V V ( ) i λ I -V + T <T> <V> ) Fo an altenative poof of the Viial Theoem see Ostlie and Caoll,pp 53-56

36 Gavitational Collapse Imagine a cloud mass M, unifom density ρ, adius o M 4 3 π 3 o ρ held at o and eleased In the absence of othe foces, cloud will collapse Consevation of enegy GM GM o ˆ t ff t ff I 0 dt o I 0 Š Œ dt d d o I 0 GM GM o / d Substitute x / o

37 4-76 t ff Š Œ 3 o / I GM 0 Š Œ x x / dx Put x sin θ; integal is π/ t ff Š Œ 3 π 3G ρ Collapse time is independent of initial size Fo Sun, ρ4 gm cm -3 t ff 8 x 0 3 sec 30 minutes /, depending only on ρ