The Virial Theorem

…and maybe a bit on timescales

Rudolf Clausius (1822-1888)

• First and second laws of thermodynamics (equivalence of heat & work; heat flows from hot to cold), 1850 • Kinetic theory of gases, 1857 (Über die Art der Bewegung welche wir Wärme nennen) • Concept of entropy, 1865 • Virial theorem, 1870

1 Translation vs. Rotation v w Translation Rotation r v † r Position r q Angular position † r Velocity vr w Angular velocity r r Acceleration a a Angular acceleration Mass m I Moment of inertia

ous † † † †

Moment of inertia (tröghetsmoment)

M 2 continuous mass distribution: I = Ú dm r 0 2 discrete masses: I = Â mi ri i

†

†

2 Analog quantities v w Translation Rotation r v † r Momentum pr = mvr L = Iwr Angular momentum † r r r r Force F = ma t = Ia Torque Kinetic K = 1 mv2 K = 1 Iw2 Kinetic energy †2 2 energy † † † † †

Moment of Inertia: Time Derivatives

2 r r I = Â mir = Â miri ⋅ ri i i dI Â r˙ r = 2 mi ri ⋅ ri the "virial" dt i † 2 d I 2 r r 2 = 2 Â mi r˙i + 2 Â Fi ⋅ ri dt i i †

† 3 The Virial of Clausius

r r Ê 1 r r 1 r r ˆ  Fi ⋅ ri = ÂÁ 2  Fij ⋅ ri + 2  Fij ⋅ ri ˜ i i Ë j≠i j≠i ¯ r r Fi =  Fij j≠i r r 1 r r 1 r r  Fi ⋅ ri = 2   Fij ⋅ ri + 2   Fji ⋅ r j i i j≠i j i≠ j r r † F = -F † ij ji r r  F r 1   F r rr † i ⋅ i = 2 ij ⋅( i - j ) † i i j≠i

† The virial for a self-gravitating system… r r  r 1   r r Fi ⋅ ri = 2 Fij ⋅(ri - rj ) i i j≠i r Gmim j r r Fij = - 3 (ri - r j ) r - r i j Gm m † r r 1 i j  Fi ⋅ ri = - 2   = W † i i j≠i rr - r i j …is just its gravitational potential energy

†

4 The (Scalar) Virial Theorem

2 1 d I 2 r r 2 = Â mi r˙i + Â Fi ⋅ ri 2 dt i i 1 d2 I = 2 K + W 2 dt 2 †

Relation between kinetic and potential energy †

Virial Theorem (simplified form)

1 d2 I = 2 K + W 2 dt 2 T time Q lim 1 Q(t)dt average = T Æ• T Ú 0 † 2 K + W = 0 † † W = - 2 K

†

† 5 Consequences of Virial Theorem

E = K + W K W = - 2 K 0 E What happens when a system W † (such as a star) contracts?

† • Loses energy and heats up (negative heat capacity!) • Half of energy loss Æ heats the system • Half of energy loss Æ radiated away

Application: Jeans Instability

E = K + W < 0

P = nkT K < W 2 cs = P r

† † 1/2 Ê 1 ˆ† R >~ Á ˜ cs † Ë Gr0 ¯

† 6 Fritz Zwicky (1898-1974)

• Swiss astronomer who spent many years at Caltech • Studied the distribution of galaxies in Coma Berenices, 1933 • Insight into nature of extragalactic supernovae; coined the term "neutron star" • "Supernovae and Cosmic Rays," Zwicky & Baade, 1934 • First to consider gravitational lensing by extragalactic objects, 1937

Masses of clusters

7 Application: Coma Cluster

• Cluster of > 1000 galaxies • Mean redshift: 7000 km/s • Mean radius (at which projected surface density falls to half of peak value) is 9' on the sky. This is about 0.4 the gravitational radius. • Dispersion in radial velocity is about 1020 km/s. • Like Zwicky, we can use observations + virial theorem to "weigh" the Coma cluster.

IGM and Dark Matter

Optical image X-ray (Chandra) image

8 Thoughts on the Virial Theorem

• What did we leave out? • How about different masses? • Collisions? Close encounters? • Timescales

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