The First Evidence for Dark Matter: The Virial Theorem and Galaxy Cluster Motion Prof. Luke A. Corwin PHYS 792 South Dakota School of Mines & Technology Jan. 16, 2014 (W1-2) L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 1 / 18 Outline 1 A little History 2 The Virial Theorem 3 Application to Galaxy Clusters 4 Reminders L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 2 / 18 A little History Fritz Zwicky \If ever a competition were held for the most unrecognized genius of twentieth century astronomy, the winner surely would be Fritz Zwicky. A bold and visionary scientist, Zwicky was far ahead of his time in conceiving of supernovas, neutron stars, dark matter, and gravitational lenses. His innovative work in any one of these areas would have brought fame and honors to a scientist with a more conventional personality. But Zwicky was anything but conventional. He once said, `Astronomers are spherical bastards. No matter how you look at them they are just bastards.' His colleagues did not appreciate this aggressive attitude and, mainly for that reason, despite Zwicky's major contributions to astronomy, he remains virtually unknown to the public."1 1Steven Soter and Neil deGrasse Tyson, Eds., Cosmic Horizons: Astronomy at the Cutting Edge. New Press, 2001. L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 3 / 18 A little History L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 4 / 18 A little History Zwicky's Work Figure : Zwicky was measuring the redshifts and distances to various closers of galaxies. This work followed on Hubble and Slipher's work and was part of establishing the expansion of the Univerise. First reported in Helv. Phys. Acta 6 (1933) 110-127 L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 5 / 18 A little History An Anomaly Figure : The average velocity of each cluster fit well with what would come to be known as Hubble's Law, but the range of velocities of the galaxies within the cluster was much larger than expected. To understand the significance of this, we need the Virial Theorem. L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 6 / 18 The Virial Theorem Newton's Law of Gravitation At this level General Relativity is not yet required. As you know, equating Newton's Second Law with this Law of Gravitation yields Gm m F = m a = i j ; (1) i r2 where mi is the mass of the object in which we are interested, and a its its acceleration. mj is any other point mass that is gravitationally affecting it and r2 is the distance between them. L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 7 / 18 The Virial Theorem Once Again, With Vectors! We can rewrite Equation 1 in terms of a position vectors ~r ¨ Gmimj F = mi~ri = 2 ; (2) j~ri − ~rjj where we have rewritten the acceleration as d2 a = ~r (t) ≡ ~r¨ (3) i dt2 i i L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 8 / 18 The Virial Theorem Mathematical Manipulation If we take the dot product of both sides with ~ri, we obtain ¨ ~ri · (~ri − ~rj) F = mi~ri · ~ri = Gmimj 3 ; (4) j~ri − ~rjj We will be dealing with systems that contain larger numbers of \particles." To account for the total force felt by the particle i from all other particles in the system, we sum over them ¨ X ~ri · (~ri − ~rj) mi~ri · ~ri = Gmimj 3 ; (5) j~ri − ~rjj j6=i L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 9 / 18 The Virial Theorem By appropriately using the time derivatives of ~ri (see the homework), we obtain 1 d2 ~r · (~r − ~r ) 2 _ 2 X i i j 2 (mi~ri ) − mi~ri = Gmimj 3 (6) 2 dt j~ri − ~rjj j6=i Since we are interested in the dynamics (e.g. the potential and kinetic energy) of the entire system, we sum over all particles i 1 d2 ~r · (~r − ~r ) X 2 X _ 2 X X i i j 2 (mi~ri ) − mi~ri = Gmimj 3 (7) 2 dt j~ri − ~rjj i i i j6=i L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 10 / 18 The Virial Theorem The RHS of can be rewritten (again, see homework) to obtain 1 d2 1 Gm m X 2 X _ 2 X i j 2 (mi~ri ) − mi~ri = − (8) 2 dt 2 j~ri − ~rjj i i i;j;j6=i The RHS is now just the gravitational potential energy of the system U _ Note that ~ri is just the velocity ~v and thus X _ 2 X 2 − mi~ri = − mivi = −2T; (9) i i where T is the total (classical) kinetic energy of the system. L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 11 / 18 The Virial Theorem Obtaining the Theorem 1 d2 X (m ~r 2) − 2T = U (10) 2 dt2 i i i If the system is in equilibrium 1 d2 X (m ~r 2) = 0 (11) 2 dt2 i i i 1 T = jUj (12) 2 L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 12 / 18 Application to Galaxy Clusters Baryonic Mass of the Coma Cluster We can count the galaxies and estimate the total mass as Zwicky did: 42 11 M ∼ 1:6 × 10 kg = 8:0 × 10 M (13) Since Zwicky's time, more mass has been discovered in the cluster in the form of hot gas between and within the galaxies, for a modern total mass estimate2 of −1 −5=2 13 M = (1:0 ± 0:2)h + (5:45 ± 0:98)h × 10 M (14) Using Hubble scale parameter h = 0:673 ± 0:012 (from PDG): 14 M = 1:6 × 10 M (15) 2Nature 366 (1993) 429 L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 13 / 18 Application to Galaxy Clusters Calculating Energies For a spherically symmetric distribution of galaxies, the total potential energy is GM 2 jUj = ; (16) R where R is the radius of the cluster. Via redshift, we can only measure velocities in parallel with the line of sight (vk). If we assume velocities are distributed 2 2 isotropically, the average velocity hv i = 3hvki. Thus, the total kinetic energy of the cluster is 3 T = Mhv2i (17) 2 k L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 14 / 18 Application to Galaxy Clusters Total Mass of the Coma Cluster Combining Equations 16 and 17 via the Virial Theorem yields 3 GM 2 3Rhv2i Mhv2i = ) M = k (18) 2 k 2R G q 2 hvki = 1008 km=s Astrophys. J. 125 (1999), 35 −11 m3 G = 6:67388(8) × 10 kg·s2 PDG R = 2:2 Mpc = 8:4 × 1022 m Nature 366 (1993), 429 & PDG 15 M = 1:9 × 10 M (19) Compare that with the visible total from a few slides ago: 14 M = 1:6 × 10 M (20) L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 15 / 18 Application to Galaxy Clusters Conclusions Unchanged Since 1933. Zwicky (translated from the original German) \In order to obtain the observed value of an average Doppler effect of 1000 km/s or more, the average density in the Coma system would have to be at least 400 times larger than that derived on the grounds of observations of luminous matter. If this would be confirmed we would get the surprising result that dark matter is present in much greater amount than luminous matter." More recent observations have reduced the discrepancy from a factor of 400 to an order of magnitude, but the dark matter remains. L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 16 / 18 Application to Galaxy Clusters Looking Forward Zwicky (translated from the original German) \These considerations show that the great dispersion of velocities in the Coma system (and in other dense nebular clusters) harbours a problem that is not yet understood." Amazingly, the final phrase still applies 80 years later, but much progress has been made In coming classes, we will discuss other evidence for dark matter, but this is how the mystery began We may be the first generation to solve this mystery that was articulated more than 80 years ago! L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 17 / 18 Reminders Reminders First Homework Due Jan. 21 Choose your topic for mid-term presentation before Jan. 30 Choose your topic for final presentation on or before Feb. 20 L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 18 / 18.