• We can use Gauss’ theorem and the full apparatus of potential theory to solve gravitational problems.
• In terms of the potential we have Poisson’s equation: • All material bodies attract one another: force proportional to the
∇2 φ = 4πGρ
product of masses; inversely proportional to square of separation.
• Force on a body of mass m at position r is determined by an acceleration field g(r): F = mg . This gravitational field is the sum of X Gmi (ri − r) accelerations from all bodies: g(r) = |ri − r|3 i • Gravitational field is conservative — derived from potential φ(r): X Gmi g = −∇φ; φ(r) = − |r i − r| i q GM • Analogy with electrostatics: φel = . ↔φ=− 4π0 R R
• We will only need cases where there is high symmetry: - spherical distribution gr inside r ;
=
GM (r) , where M (r) is the mass r2
- cylindrical distributions will not arise in this part of the course. - symmetric slab distribution gz
= 2πGM (z) (simple model of a
gravitating disc).
• In fact we can do quite complicated problems. . .
• Consider volume V surrounded by surface S • Gauss’ theorem in electrostatics: I Z Z Q 1 ρel dS·E = dV ∇·E = dV ρel ⇒ ∇·E = = 0 0 0 • For gravity the analogy gives I Z Z dS·g = dV ∇·g = −4πGM = −4πG dV ρ ⇒ ∇·g = −4πGρ
Gravitational fieldlines and equipotentials for a uniform disc.
Newtonian Gravity
P3
Newtonian Gravity
P4
DYNAMICS OF EXPANDING SHELL GRAVITATIONAL FIELD AND POTENTIAL OF A SPHERICAL BODY
• Spherical body of density ρ and radius a (total mass M = 43 πρa3 ). • Field at radius r determined by mass M (r) inside: g(r) =
GM (r) . r2
• Outside the body we just have: GM GM g(r) = 2 ; φ(r) = − . r r • Inside the body M (r) =
M r3 , a3
GM r = 43 πGρr. so that g(r) = a3
• The potential φ(r) is the integral of g(r) with the constant of integration chosen to match the potential at r = a: GM 2 r − 3a2 . φ(r) = 3 2a
• If the matter in the sphere is at rest and has the correct pressure distribution P (r) it can support itself against collapse. • Hydrostatic equilibrium: balance of forces on element ∇P = ρg . Here ρ is constant so we can integrate: P = −ρφ + constant. • Setting P = 0 at r = a gives P (r) = 23 πGρ2 a2 − r2 3GM 2 • In terms of mass and radius the central pressure is P0 = 8πa4
• If there is no pressure (a “dust” sphere), a uniform sphere started at
rest will collapse (see examples1: Q6 — method given below works).
• The gravitational field is proportional to radius, so that the collapse has radial velocity v(r) ∝ r , and the density remains spatially uniform. • We call this type of motion “self-similar‘’, and we will meet it several times (outflows; blast waves).
• We will look at the case of an expanding sphere with v(r) ∝ r, and use it as a simple Newtonian model for the expanding universe.
• We write the radial position r(t) of a particle in the sphere as λR(t), where 0 < λ < 1 is a time-invariant label and R(t) is the radial scale a3 . (R(t = 0) = a). We also have ρ(t) = ρ0 3 R (t) ˙ . • Then v(r, t) = λR˙ = H0 (t)r, where H0 ≡ R/R • With these substitutions the equation v˙ = −g becomes (λ cancels): 3 ¨ = − 4πGρ0 a R 3R2 4πGρ0 a3 + constant. • Multiply by R˙ and integrate: 12 R˙ 2 = 3R • The constant of integration determines whether the expansion will
cease: if negative the sphere stops and recontracts. The critical case
3H02 (t = 0) ˙ . is has R = 0 at R = ∞, so ρcrit = 8πG • Not a bad model for the universe. . .
Newtonian Gravity
P5
Newtonian Gravity
P6
VIRIAL THEOREM
THE FRIEDMANN EQUATION
• Our expanding dust model with Newtonian gravity is actually a very
• Balance between gravitational potential energy and kinetic energy in
systems bound by gravity and supported against collapse by internal
good model for the expanding universe — it has reproduced the
motions.
Friedmann equation for the evolution of the cosmological scale parameter. It is the same equation you get from general relativity if the pressure and cosmological constant are zero.
• By appropriate scalings of time and radius it can be put in the form 1 R˙ 2 = − k , where distinct cases are distinguished by k taking one R of the three values −1, 0, +1.
• System of masses mi , positions r i and velocities v i X • The quantity G ≡ (mi v i )·ri is called the virial. i
X dG X mi v 2i + v˙ i ·r i = 2T + Fi ·r i = dt i i
