Lecture 2 Key Concepts Newtonian echanics and relation to Kepler's laws The Virial Theore Tidal forces Collision physics Newton's Laws 1) An object at rest will reain at rest and an object in otion will reain in otion in a straight line at a constant speed unless acted upon by an external force (Conservation of linear oentu in absence of unbalanced force). 2) Force is the change of oentu with tie. 3) The law of action-reaction, or conservation of total linear oentu. 1 2 Physical interpretation of Kepler's Laws 1. "A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse". With Newton, the orbit shape depends on E, which is the su of each body s KE plus the PE: The center of ass is always on a line between two objects: CM 2 Total energy E < 0 circular or elliptical (bound) = 0 parabolic (arginal) > 0 hyperbolic (unbound) 1 C&O show that center of ass is unaccelerated if no external forces exist. Define the displaceent vector For two asses in bound orbit, orbits are ellipses with the center of ass at the coon focus. Orbits easier to understand in center of ass reference frae. Define reduced ass, and the total ass, as 3 4 Can easily show: they are antiparallel. Note: r 1 + r 2 =r since Kepler's Second Law Revisited "A line connecting a planet to the Sun sweeps out equal areas in equal tie intervals" Center of ass frae very convenient for total energy (KE+PE): (Proble 2.4) Consider area of ellipse in polar coordinates da dr r where Also, angular oentu (Proble 2.5) So, integrating over r, area fro focus to distance r is as orbits CM, it sweeps out this area focus => Orbit is equivalent to reduced ass orbiting non-accelerated ass M at a distance r (with sei-ajor axis a) and speed v. Can also show a = a 1 + a 2 (Proble 7.1). C+O show the orbit of is a conic section. 5 6 1
The orbital velocity v can be decoposed: Kepler's third law: It can be shown that the total orbital angular oentu for closed orbits is (understand this in C+O!): Q: For what shape orbit is L at its axiu? Miniu? Now over is therefore constant, so integrating: Also, for an ellipse (see prob 2.2.) Substitute, square and re-arrange: => Area swept out at constant rate! 7 8 Using expressions for L above and recalling fro lecture 1 that Newton's for of Kepler's third law The Virial Theore For an isolated, gravitationally bound syste in equilibriu, the total energy (constant) is always one-half the tie-averaged potential energy. Q s: Why did Kepler iss the ( 1 + 2 )? Since E = <U> + <KE>, 2<KE> + <U> = 0 What systes would you observe where you couldn't iss it? How would you go about weighing stars? This is extreely useful, applies to a wide variety of astrophysical situations, whether you have two objects or a large nuber. 9 10 Exaple: How to weigh a cluster of galaxies? Assue a spherical cluster with N galaxies, with tie-average separation R, and rando tie-average speed <V>, individual galaxy ass. Note that if in equilibriu, we can assue tie average = enseble average, which we can easure. Since N>>1 Total ass Nuber of pairs: N(N-1)/2 Exaple: We see a velocity dispersion of 670 k/s in the Virgo Cluster which is 3 Mpc in diaeter and roughly spherical. What is the ass of the Cluster? 11 12 2
Tidal forces Collision physics Lecture 3 Key Concepts Tidal forces (Sec 19.2) Differential gravitational force = difference between the gravitational forces exerted on two neighboring particles by a third, ore distant, body M r dr 13 14 Tides arise because of the differential gravitational force on opposite sides of the. Not quite the whole story though: Fro a "center of " perspective, subtract fro each vector: Stretching to - axis Squeezing to - axis Net effect is that for to bulge both towards and away fro the. See C&O 19.2 for a full-blown treatent. Quantitatively, stretching (df/dr) is a larger ter. Qualitatively, reeber both - black holes. Q: Why is the 's orbital otion deterined by the Sun, but the 's tides are generated principally by the? 15 16 Spin and orbit evolution: 's spin angular oentu changes: Friction => tidal bulge leads - line Also, dragged along faster, igrates outward 4 c/year, gains angular oentu. Total of syste is conserved. Q: What would happen if the satellite were leading the planet's tidal bulge? has a coponent opposing 's rotation (torque). 17 18 3
Final point about tidal forces - The Roche Liit: The distance fro a planet below which the differential pull of the planet on two neighboring particles exceeds their utual gravitation. R d d >> r 2r M p Consider two sall asses, each of ass, radius r, at a distance d fro center of the planet with ass M p. At the Roche Liit: differential attraction by planet = utual attraction of two asses. 19 More precise derivation (one spherical body instead of two spheres) gives sae but constant is 2.44. For gravitationally bound lups inside Roche Liit, breakup due to tidal forces will occur. 20 Caution: Satellites, people etc., can live inside Roche liit - why aren't we being pulled to pieces? Exaple: Coet Shoeaker Levy 9. Passed inside Jupiter s Roche Liit on previous passage. How close would it have to get? Another iplication is that the Roche Liit is an accretion liit, outside of which clups can becoe gravitationally bound, and inside of which tidal forces will disrupt the. Coet nuclear density ~ 0.5 g c -3. Jupiter s density 1.33 g c -3. Radius 74,500 k. d ~ 250,000 k 21 22 Collision physics A very iportant and powerful eans of aking quantitative estiates about things with siple arithetic. You can answer questions like: How long before a ajor city is destroyed by a collision with an asteroid? Do stars collide when galaxies collide? How far does an ato travel between collisions deep in the solar interior? Assue: A cloud of identical particles, each with radius r and cross section σ=πr 2 Mean nuber density of particles is n (#/volue) Typical velocity with respect to cloud of particles is v. Then in a tie t, a particle oving with velocity v sweeps out a volue σvt. Nuber of particles in that volue it will collide with is Distance traveled between collisions is Tie between collisions is 23 λ is ean free path τ is ean free tie 24 4
τ is the ean free tie a given particle spends between collisions - not the average tie between any collisions in the entire cloud of particles. Will apply this to how photons ove through stellar interiors and atospheres later. An extree case of tides: picture an iron cube, 1c on a side held just above a neutron star R=10 k, M=1.4 M, density iron 7.86 g c -3. Assue you divide the cube into two halves, with half of the ass in each part. The tidal stress on the cube can then be estiated as: Exaple: How long until the Sun collides with another star? How does this copare to the age of the Universe? How likely is it that the Sun will collide with another star? 25 An iron eteorite falling toward the surface of a neutron star would be stretched (to the point of rupturing) into a thin ribbon. 26 5