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 =  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 =  mir =  miri ⋅ ri i i dI  r˙ r = 2 mi ri ⋅ ri the "virial" dt i † 2 d I 2 r r 2 = 2  mi r˙i + 2  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 † r Gmim j  F ⋅ r = - 1   = W † i i 2 r r i i j≠i ri - rj …is just its gravitational potential energy † 4 The (Scalar) Virial Theorem 2 1 d I 2 r r 2 =  mi r˙i +  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 9.