Physics 312 Intoduction to Astophysics Lectue 7 James Buckley buckley@wuphys.wustl.edu Lectue 7 Eath/Moon System Tidal Foces Tides M= mass of moon o sun F 1 = GMm 2 F 2 = GMm ( + ) 2 Diffeence in gavitational foce on nea and fa side esults in oceans being pulled moe stongly than aveage gavitational foce on the igid body of the eath on nea side, moe weakly on fa side
Tides Note that the adius of the eath = 12, 742km is much smalle than the distance between the eath and moon moon = 384, 000km and the distance between the Eath and the sun sun = 149, 000, 000 km = 12 384 0.03 F 2 = GM m 2 (1 + ) 2 2 =(0.03) 2 =0.0009 GM m 0 = 2 (1 + 2 + 2 ) GM m 1 2 2 (1 + 2 ) 1 2 GM m 1 2 0 2 1 4 2 GM m(1 2 ) 2 (define /) F 1 = GMm 2 F 2 = GMm ( + ) 2 Sum of foces F 1 + F 2 + F othe stu combines to keep Eath in cicula obit. Di eence, stetches oceans and makes tide Stetching, tidal foce = F 1 F 2 Tidal foce = 2GM m GMm Tidal foce = 2 [1 (1 2 )] = 2GM 3 2 Sping Tides Note that on the nea side, thee is a slightly lage foce than needed to keep the igid planet in obit aound the sun, but on the fa side slightly less - esulting in an effective stetching foce on both sides. So even when it is full moon, and it appeas that the foces of sun and moon wok against each othe, both povide a positive effective stetching foce, adding the tides. At full and new moon, we get stonge Sping tides
Neap Tides At fist and thid quate moon, effects of Sun and moon wok against each othe poducing weake neap tides Tidal Locking Nea Side (we always see this face fom Eath) Nea Side (we always see this face fom Eath) Moon is tidally locked to the Eath, we always see the same side (left) neve the fa side (ight). How did this come to be? If the moon wee not tidally locked, it would be compessed one way, then the next. All of this cunching dissipates enegy and the moon is only happy when it is locked in place!
Tidal Disuption Conside the tidal foce of a lage body of mass M on a smalle body, modeled vey oughly as two smalle masses m sepaated by a distance M m m Taylo seies expansion F () F ( 0 )+( 0 ) df ) d =0 F () = GMm df ) 2 dr = 2GMm 3 F = df d We can calculate the Roche limit the point at which the tidal foce ipping the two masses apat is geate than the self-gavity that pulls the masses togethe: 2GMm R 3 = Gmm 1/3 2M ( ) 2 ) R = m Rewiting in tems of density: M = 4 3 R3 M, m = 4 3 1/3 M m ) R 2.5 R 3 2 m 1/3 M Which is vey close to the moe caeful esult: Roche limit: R =2.44 R m Discussion Question Would you be less likely to find a Jovian (gas giant) planet o a teestial (ocky) planet vey close to anothe sta? If the adius of the event hoizon of a black hole is popotional to the mass of the black hole, would you be moe likely to suvive cossing the hoizon of a sola mass black hole o supemassive black hole?
Spaghettification { F ( + ) = GMm ( + ) 2 F () = GMm 2 F = GMm 1 1 2 ( + ) 2 F 2GMm 3 At the event hoizon = sch = 2GM c 2 So the tidal foce at the event hoizon is : F 0.4 10 12 lb 2 Msun M bh m astonaut 220 lb height 6 ft A black hole with the mass of the sun would snap you in half with a foce of half a tillion pounds at the event hoizon! Black Hole Oh Snap! of mass Mbh need to find a million-sola-mass black hole to suvive the jouney! Toques in Eath Moon system (1) Fiction in otating Eath pulls tidal bulge slightly ahead of moon (3) The Eath s bulge pulls the moon ahead, inceasing its obital distance (2) The moon gavity ties to pull the Eath s bulge back, slowing the otation Time out! How do we know that the tug of the eath doesn t incease the velocity, and decease the adius fom Keple s laws? If the adius inceases does the velocity decease?!
Angula Momentum Makes it Clea ~L = ~ ~p ~v ~L = ~ (m~v) ~L = ~ (m~v) =mv! ~ v 2 = GM GM v = p p L = GM m As inceases, the angula momentum inceases (fo a given mass m) So if angula momentum is conseved, and the Eath s angula momentum deceases, the angula momentum of the Moon s obit must incease, so inceases and v deceases! Equatoial Bulge In addition to tidal foces, the spin of the eath esults in an equatoial bulge (Kind of like the middle age spead o love handles fo planets)
Pecession 23.5 plane of the ecliptic F 1 = GMm 2 F 2 = GMm ( + ) 2 Diffeence in gavitational foce on nea and fa side bulge poduces toque, tying to twist the planet. But like a gyoscope, the toque doesn t twist the spinning planet the way we would expect, but instead changes the diection of the angula momentum in the diection of the toque (see demonstation) Pecession Angula Momentum L Gavitational Foce F Gavitational foce ties to twist the top, with a toque whose diection is pependicula to both the axis and the foce The angula momentum vecto changes its diection in the diection of the toque
Pecession Eath pecesses due to toque fom gavitational foces of the moon and sun on the equitoial bulge 26,000 yea peiod ove which pole sta changes, it will not always be Polais, o any sta fo that matte. Pecession Pecession is the slow (25,770 yea peiod) wobble of the eath s inclined obit. RA and DEC given in an epoch (e.g., 1950) and must be pecessed to cuent time. α = [m + n sin α tan δ] N δ = [n cos α] N N numbe of yeas fom efeence epoch 1950.0 m = 3.07327 s y 1 and n = 20.0426"y 1 Physics 312 - Lectue 5 p.9/12
Spookily Simila - Moon and Sun The fact that the angula size of the moon and sun means that the effect of these two objects on the Eath. Stat with Newton s law of Gavity: The gavitational foce between two masses is popotional to the poduct of the masses and invesely popotional to the squae of the distances between the centes of the two masses M 1 + + M 2 F = GM 1 M 2 2 Tidal Foces But the foce of gavity on the nea side of object M2 is lage than on the fa side. The diffeence in these foces is called the Tidal foce, and tuns out to be popotional to the poduct of the two masses and the atio of the length of mass M2 divided by the cube of the distance between the two objects + M 1 + + M 2 F nea Stetching o tidal foce = F nea F fa 1 1 = GM 1 M 2 2 ( + ) 2 if is small compaed with, you can show that F tidal GM 1 M 2 3 F fa
Compae the Sun and the Moon Now, let s compae the effect of the Sun and the Moon on the tidal foces on the eath: Aveage density of the Sun is 1.408 g/cm 3 and of the moon is 3.3 g/cm 3 - oddly simila! The sun and the moon have about the same angula size, 0.5 deg Tidal foces ae popotional to 1/d 3, The mass of the moon o sun is popotional to the poduct of the density times the volume. The volume of a sphee is popotional to the adius cubed, and mass is adius times volume so: Since the angula size of an object is oughly the atio of it s physical dimension (2 R) divided by the distance, and since the moon and the sun have the same angula size: So, compaing the tidal foces: M sun / 1.4 R 3 sun, M moon / 3.3 R 3 moon moon sun 0.5 =2R sun /d sun =2R moon /d moon F moon M eath (3.3 R 3 moon) d 3 moon F sun M eath (1.4 R 3 sun) d 3 sun so since R moon /d moon R sun /d sun we have : F moon (3.3/1.4)F sun The Sun and the Moon The sun and the moon have about the same angula size and close to the same aveage density. Thei impact on pecession and the tides ae quite simila! Because the moon just blocks the sun duing an eclipse, allowing humans to see the oute layes of the sun - coona and flaes! A fotuitous occuence! The moon doesn t shine on its own, but is eflected light fom the sun. Since it is petty gay, it is not eflecting all of the light (but it still looks white at night against the black sky)