Dark Matter and Individual Galaxies

Dark Matter and Individual Galaxies

Prof. Luke A. CorwinPHYS 792

South Dakota School of Mines & Technology

Jan. 16, 2014 (W2-1)

L. Corwin, PHYS 792 (SDSM&T) Galactic Motion Jan. 16, 2014 (W2-1) 1 / 18

Outline

1 Kepler & Newton

2 Vera Rubin & Kent Ford

3 Satellite Galaxies

4 Reminders


Kepler & Newton

Today, we turn to probably the most famous evidence for darkmatter: galactic rotation curves. In the case of a star in a circularorbit around a spherically symmetric galaxy, we can set thegravitation acceleration due to the mass M(≤ r) inside of theorbital radius r of the star equal to the centripetal acceleration:

GM(≤ r)

r2=v2(r)

r⇒ v(r) =

√GM(≤ r)

r(1)

We can thus determine the mass of a galaxy with a radius r bymeasuring the velocities of stars at r. This still works for ellipticalorbits; the equations are just more complicated.


Kepler & Newton

Another Implication

GM(≤ r)

r2=v2(r)

r⇒ 1

v2(r)=

r

GM(≤ r)(2)

Note that the orbital period P = (2πr)/v(r)

P 2 =

(2πr

v(r)

)2

=(2π)2

GM(≤ r)r3 (3)

⇒ P 2 ∝ r3 (4)


Kepler & Newton

Question for Class: DoesEquation 4 look familiar?


Kepler & Newton

That equation may look familiar.It is the special case of Kepler’sThird Law

P 2 ∝ a3, (5)

where a is the semi-major axis.a = r for a circular orbit.(http://xkcd.com/21/)


Kepler & Newton

Mass Distribution (In-Class Exercise)

1 Supposing all of a galaxy’s mass is in luminous matter andvisible gas, beyond the visible limits of the galaxy, M(≤ r)becomes constant. What velocity distribution would oneexpect near the edge of a galaxy in that case?

2 As you may know, the the actual velocity distributions ofmany galaxies are flat in their outer regions (v(r) ∝ r). Whatmass distribution does that imply?


Kepler & Newton

Answers

1.) If M(≤ r) is constant,

v(r) =

√GM(≤ r)

r∝ 1

r, (6)

so the velocity distribution would fall of as 1/r.

2.) Rearranging yields

M(≤ r) =v2(r)r

G(7)

If v(r) is constant, then M ∝ r.


Vera Rubin & Kent Ford

Figure : Dr. Vera Rubin and Kent Ford established the anomalousrotation curves of multiple spiral galaxies in the early 1970’s2.

2Astrophysical Journal 238, 471; “Rubin, Vera Cooper” in Lisa Yount, Ato Z of Women in Science and Math Facts on File (2007); Photo: EmilioSegre Visual Archives



Galactic Rotation Curves

One of the iconicgalactic rotationcurves from Mon.Not. Roy. Astron.Soc. 249 (1991)523. “. . . thedashed curves arefor the visiblecomponents, thedotted curves forthe gas, and thedash-dot curves forthe dark halo.”



Important Points

The flat rotation curves of galaxies remains a persistent,powerful, and easy to understand piece of evidence for theexistence of dark matter.

Every galaxy for which measurements exist shows thisevidence for dark matter with ratios of mass to luminosity≥ 10.

One major discriminating characteristic between small dwarfspheroidal galaxies and globular clusters is that globularclusters are not dominated by dark matter, but the galaxiesare.


Satellite Galaxies

Dark Matter in Milky Way Satellites3

Figure : Left: Bootes I (Sloan Digital Sky Survey) Right: LargeMagellenic Cloud (NASA); two satellite galaxies of the Milky Way.

3Most of this section taken from Ch. 3 of G. Bertone, editor, ParticleDark Matter: Observations, Models and Searches, Cambridge UniversityPress: Cambridge, UK (2010)


Satellite Galaxies

Satellite Galaxy Characteristics

Our home galaxy, the Milky Way, has at least 24 knownsatellite galaxies; the majority were discovered by the SloanDigital Sky Survey (SDSS) since 2004.

They have been the subject of active study with astronomersattempting to determine, among other things, their darkmatter content.

Difficult to detect behind the foreground Milky Way stars

Distances from the sun in the range 25− 420 kpc.

Sizes (half-light radii) in the range 10− 400 pc.

Since the SDSS only surveyed 20% of the sky, many more areprobably undiscovered


Satellite Galaxies

Tidal Forces

When modeling the motions of stars in satellite galaxies, one mustconsider the tidal forces exerted by the Milky Way. A simpleestimate of the internal gravitational force on stars in the outerparts of the galaxy is

σ2r

Rs

, (8)

where σ2r is the radial velocity dispersion of the stars in the galaxy

and Rs is its distance from the center of the Milky Way. The tidalforce from the Milky Way is on the order of(

∼ 220km

s

)2Rs

D, (9)

where D is the distance of the satellite from the center of theMilky Way


Satellite Galaxies

Tidal Forces

For the known satellites σ2r ≈ 5− 15 km/s

Rs ≈ 10− 400 pc

Data from satellites in the range D ∼ 20− 250 kpc werestudied

With these values, the stars in the satellite galaxies arebound by gravitational forces ∼ 100 greater than the tidalforces in the Milky Way.

This allows us to assume the motions of the stars representthe galaxies gravitational potential

In order to make a consistent comparison, the mass within300 pc of the center of each galaxy is measured.


Satellite Galaxies

Figure : The mass of Milky Way satellites is remarkably constant4

4Nature 454:1096 [arXiv:0808.3772]L. Corwin, PHYS 792 (SDSM&T) Galactic Motion Jan. 16, 2014 (W2-1) 16 / 18

Satellite Galaxies

Implications

Across 7 orders of magnitude in luminosity, the mass ofMilky Way satellites is remarkably constant at ∼ 107M�

“This result demonstrates that the faintest of the Milky Waysatellites are the most dark matter-dominated galaxiesknown, and could be a hint of a new scale in galaxy formationor a characteristic scale for the clustering of dark matter.”


Reminders

Reminders

Choose your topic for mid-term presentation before Jan. 30

Presentations will be given (two per class period) duringWeek 7 (Feb. 25 and 27)The first person to inform me of their topic choice will havetheir choice of presentation date.

Choose your topic for final presentation on or before Feb. 20


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Dark Matter and Individual Galaxies