The First Evidence for Dark Matter: The Virial Theorem and ...odessa.phy.sdsmt.edu/~lcorwin/PHYS792DM_Spring2014/PHYS792… · contributions to astronomy, he remains virtually unknown

The First Evidence for Dark Matter:

The Virial Theorem and Galaxy Cluster Motion

Prof. Luke A. CorwinPHYS 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


A little History

Fritz Zwicky

“If ever a competition were held for the most unrecognized geniusof twentieth century astronomy, the winner surely would be FritzZwicky. . . A bold and visionary scientist, Zwicky was far ahead ofhis time in conceiving of supernovas, neutron stars, darkmatter, and gravitational lenses. His innovative work in anyone of these areas would have brought fame and honors to ascientist with a more conventional personality. But Zwicky wasanything but conventional. . . He once said, ‘Astronomers arespherical bastards. No matter how you look at them they are justbastards.’ His colleagues did not appreciate this aggressiveattitude and, mainly for that reason, despite Zwicky’s majorcontributions to astronomy, he remains virtually unknown to thepublic.”1

1Steven Soter and Neil deGrasse Tyson, Eds., Cosmic Horizons:Astronomy at the Cutting Edge. New Press, 2001.


A little History


A little History

Zwicky’s Work

Figure : Zwicky was measuring the redshifts and distances to variousclosers of galaxies. This work followed on Hubble and Slipher’s workand was part of establishing the expansion of the Univerise. Firstreported in Helv. Phys. Acta 6 (1933) 110-127


A little History

An Anomaly

Figure : The average velocity of each cluster fit well with what wouldcome to be known as Hubble’s Law, but the range of velocities of thegalaxies within the cluster was much larger than expected. Tounderstand the significance of this, we need the Virial Theorem.


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 Gravitationyields

F = mia =Gmimj

r2, (1)

where mi is the mass of the object in which we are interested, anda its its acceleration. mj is any other point mass that isgravitationally affecting it and r2 is the distance between them.


The Virial Theorem

Once Again, With Vectors!

We can rewrite Equation 1 in terms of a position vectors ~r

F = mi~̈ri =Gmimj

|~ri − ~rj|2, (2)

where we have rewritten the acceleration as

ai =d2

dt2~ri(t) ≡ ~̈ri (3)


The Virial Theorem

Mathematical Manipulation

If we take the dot product of both sides with ~ri, we obtain

F = mi~ri · ~̈ri = Gmimj~ri · (~ri − ~rj)

|~ri − ~rj|3, (4)

We will be dealing with systems that contain larger numbers of“particles.” To account for the total force felt by the particle ifrom all other particles in the system, we sum over them

mi~ri · ~̈ri =∑j 6=i

Gmimj~ri · (~ri − ~rj)

|~ri − ~rj|3, (5)


The Virial Theorem

By appropriately using the time derivatives of ~ri (see thehomework), we obtain

1

2

d2

dt2(mi~r

2i )−mi~̇r

2i =

∑j 6=i


|~ri − ~rj|3(6)

Since we are interested in the dynamics (e.g. the potential andkinetic energy) of the entire system, we sum over all particles i

∑i

1

2

d2

dt2(mi~r

2i )−

∑i

mi~̇r2

i =∑i

∑j 6=i


|~ri − ~rj|3(7)


The Virial Theorem

The RHS of can be rewritten (again, see homework) to obtain

1

2

d2

dt2

∑i

(mi~r2

i )−∑i

mi~̇r2

i = −1

2

∑i,j,j 6=i

Gmimj

|~ri − ~rj|(8)

The RHS is now just the gravitational potential energy of thesystem UNote that ~̇ri is just the velocity ~v and thus

−∑i

mi~̇r2

i = −∑i

miv2i = −2T, (9)

where T is the total (classical) kinetic energy of the system.


The Virial Theorem

Obtaining the Theorem

1

2

d2

dt2

∑i

(mi~r2

i )− 2T = U (10)

If the system is in equilibrium

1

2

d2

dt2

∑i

(mi~r2

i ) = 0 (11)

T =1

2|U | (12)


Application to Galaxy Clusters

Baryonic Mass of the Coma Cluster

We can count the galaxies and estimate the total mass as Zwickydid:

M ∼ 1.6× 1042 kg = 8.0× 1011M� (13)

Since Zwicky’s time, more mass has been discovered in the clusterin the form of hot gas between and within the galaxies, for amodern total mass estimate2 of

M =((1.0± 0.2)h−1 + (5.45± 0.98)h−5/2

)× 1013M� (14)

Using Hubble scale parameter h = 0.673± 0.012 (from PDG):

M = 1.6× 1014M� (15)

2Nature 366 (1993) 429L. Corwin, PHYS 792 (SDSM&T) Virial Theorem Jan. 16, 2014 (W1-2) 13 / 18


Calculating Energies

For a spherically symmetric distribution of galaxies, the totalpotential energy is

|U | = GM2

R, (16)

where R is the radius of the cluster.Via redshift, we can only measure velocities in parallel with theline of sight (v‖). If we assume velocities are distributedisotropically, the average velocity 〈v2〉 = 3〈v2‖〉. Thus, the totalkinetic energy of the cluster is

T =3

2M〈v2‖〉 (17)



Total Mass of the Coma Cluster

Combining Equations 16 and 17 via the Virial Theorem yields

3

2M〈v2‖〉 =

GM2

2R⇒M =

3R〈v2‖〉G

(18)√〈v2‖〉 = 1008 km/s Astrophys. J. 125 (1999), 35

G = 6.67388(8)× 10−11 m3

kg·s2 PDG

R = 2.2 Mpc = 8.4× 1022 m Nature 366 (1993), 429 & PDG

M = 1.9× 1015M� (19)

Compare that with the visible total from a few slides ago:

M = 1.6× 1014M� (20)



Conclusions Unchanged Since 1933. . .

Zwicky (translated from the original German)

“In order to obtain the observed value of an average Doppler effectof 1000 km/s or more, the average density in the Coma systemwould have to be at least 400 times larger than that derived onthe grounds of observations of luminous matter. If this would beconfirmed we would get the surprising result that dark matter ispresent in much greater amount than luminous matter.”

More recent observations have reduced the discrepancy from afactor of 400 to an order of magnitude, but the dark matterremains.



Looking Forward

Zwicky (translated from the original German)

“These considerations show that the great dispersion of velocitiesin 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, butmuch progress has been made

In coming classes, we will discuss other evidence for darkmatter, but this is how the mystery began

We may be the first generation to solve this mystery that wasarticulated more than 80 years ago!


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


Documents

The First Evidence for Dark Matter: The Virial Theorem and ...odessa.phy.sdsmt.edu/~lcorwin/PHYS792DM_Spring2014/PHYS792… · contributions to astronomy, he remains virtually unknown