in Today Astronomy 142: the Milky Way, continued
Stellar relaxation time Virial theorem Differential rotation of the stars in the disk The local standard of rest Rotation curves and the distribution of mass The rotation curve of the Galaxy
Figure: spiral structure in the first Galactic quadrant, deduced from CO observations (Clemens, Sanders, Scoville and Solomon 1988)
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Stellar encounters: relaxation time of a stellar cluster
In order to behave like a gas, as we assumed last time, stars have to collide elastically enough times for their random kinetic energy to be shared in a thermal fashion. But stellar encounters, even distant ones, are rare. How long does it take a cluster of stars to thermalize? One characteristic time: the time between stellar elastic encounters, called the relaxation time. If a gravitationally bound cluster is a lot older than its relaxation time, then the stars will be describable as a gas (the star system has temperature, pressure, etc.). You will do part of a rough estimate of the relaxation time in next weeks Workshop. The following will get you started.
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Stellar encounters: relaxation time of a stellar cluster (continued)
Suppose a star has a gravitational sphere of influence with radius r (>>R, the radius of the star), and moves at speed v between encounters, with its sphere of influence sweeping out a cylinder as it does: v r vt If the number density of stars (stars per unit volume) is n, then there will be exactly one star in the cylinder if 1 nV = n r 2 vtc = 1 tc = Relaxation time 2 n r v
Astronomy 142 3
V = r 2 vt
Stellar encounters: relaxation time of a stellar cluster (continued)
What is the appropriate radius, r? Choose that for which the gravitational potential energy is equal in magnitude to the average stellar kinetic energy.
G 2 m 1 ~v , tc = 2 r nr v
v3 tc = 2 2 4Gmn
Done in more detail (Astronomy 232-level): for a spherical cluster with a core radius R (cf. next Workshop), it can be shown that v3 1 tc = . 2 2 2R 4G m n ln r (Not that far from our rough estimate.)
F I H K
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Stellar encounters: relaxation time of a stellar cluster (continued)
Next Workshop, you will show among other things that for such a cluster, with core radius R and typical stellar mass m, 2 G ( N 1) m Gm2 G ( N 1 ) m = v2 = and 2R r 4R Assume N >> 1 and substitute these into the expression for relaxation time: N N 2R tc = tx N v 24 ln N 24 ln 2 2 The time tx = 2 R v is called the crossing time; its the time it takes a star moving at the mean speed v to traverse the core of the cluster (diameter 2R) if it doesnt collide.
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Stellar encounters: relaxation time of a stellar cluster (continued)
tc tx N N 24 ln 2
Relaxation time only depends on the crossing time and number of particles in the cluster
Relaxation time determines the timescale of density evolution in a cluster. Small clusters have short relaxation times. Dense clusters near galaxy nuclear have short relaxation times. Globular clusters have relaxation times of about 109 years
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Dynamical Friction
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Dynamical friction( continued)
Distant stars wind up contributing more toward slowing down a massive body. Formula for timescale quite similar to that for the relaxation time --- this is because graviational collisions are important in both cases. Gravity is a long range force. Even though Gravity is weak, it dominates everything else and is important at large scales.
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The virial theorem
Consider an isolated system of particles that exert forces on each other, and suppose those forces can be described by potentials. Suppose also that the position configuration of the particles is either periodic (e.g. orbital motion) or statistically constant over time (e.g. thermal equilibrium). Then the systems average total kinetic energy K and average total potential energy U are related by
2 K = U .
We will resist the temptation to prove this here; though the proof isnt difficult, its long and complicated. It will be proven for you in PHY 235 and AST 242. (Note that the proof in IAA, page P12, is quite incomplete.)
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Integrated quantities, generalized virial therom
Rotation support vs thermal or kinetic support Disk galaxies are rotationally supported, whereas Eliptical galaxies are supported by random motion.
2 K + U + W = 0. K = total kinetic energy in random motions U = total gravitational potential energy W = total rotation energy
If anisotropy taken into account tensor virial theorem relating axis ratios to velocity ellipsoid.
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Rotation of the stellar population in the Solar Neighborhood
Averaging over the random motions, one can detect differential rotation in the disk of the galaxy, from the radial velocities of nearby stars. The rotation is differential in the sense that different radii have different angular velocities. The angular velocity decreases monotonically as radius from the Galactic center increases. Measurement of average stellar motions along the line of sight and perpendicular to the line of sight can be used to determine the local angular velocity and its derivative.
Astronomy 142
In frame of Sun
11
Rotation the of stellar population, and Oorts constants
Oorts constants, defined: r d A= 2 dr 1 d 2 B= r 2 r dr
vt
vr
d
vr
vt
(
)
Sun
whence = A B In terms of the average radial velocities and average proper motions as a function of l (Galactic longitude):
v r = Ad sin 2 vt = Ad cos 2 + Bd
B is usually obtained less directly from the statistics of 1 A / B = 1.6. random motions, with the result
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The local standard of rest
From A and B we get the average rotational motion of the Suns orbit, called the local standard of rest (LSR): = 8 10 16 radians s -1
v = 220 km s -1
r = 8.5 kpc
P = 230 106 years The solar system actually moves slightly with respect to the LSR, at about 7 km/s. From the motion of the LSR, the galaxy within r = 8.5 kpc can be weighed: 2 v r = 0.9 1011 M . M= G
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Rotation curves
The average orbits in the disk of the galaxy seem to be circular, centered on the Galactic center. A measurement of average angular velocity at any radius allows a determination of the mass within that radius of the Galactic center. Done as a function of radius: rotation curve Enables determination of enclosed mass, and in turn the density, as a function of r. Interstellar gas has far smaller random motions than stars, is widespread, and detectable throughout the galaxy; atomic (e.g. H I 21 cm) and molecular (e.g. CO 2.6 mm) lines are the best to use for determination of the Galactic rotation curve.
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Example rotation curves
Point mass, M: F= GMm r2 mv = r
2
GM v(r ) = r
Keplerian motion v decreases with increasing r
Constant density, spherically symmetric: 4 M (r ) = 0 r 3 3
Gm 4 mv 2 3 r = 2 3 0 r r 4 G 0 v(r ) = r 3
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Solid-body rotation v increases linearly with increasing r
15
Example rotation curves (continued)
2 Spherical symmetry, 1 / r density distribution:
2 r0
= 0
r
2 r
(r0 : core radius of galaxy) 2 dr =
2 r 40 r0 dr 0
M (r ) =
0 ( r ) 4 r
Gm
2 = 40 r0 r r
GmM ( r ) r2
mv 2 2 = 2 40 r0 r = r r
2 v ( r ) = 4 G 0 r0 = constant
Flat rotation curve
As we will see, many rotation curves of disk galaxies, including ours, look like this one.
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Measurement of Galaxys rotation curve from H I and CO line profiles
Wavelength or frequency shift and radial velocity: the Doppler effect.
0 vr = = = 0 0 0 c
Maximum radial velocity must come from orbit tangent to line of sight: distance and rotational motion of tangent points very well determined. Distance ambiguity: for lines of sight toward the inner galaxy (first and fourth quadrant), there are two locations with the same radial velocity.
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Measurement of Galaxys rotation curve from H I and CO line profiles (continued)
Resolution of the ambiguity usually involves information other than velocities association or lack thereof with visible-wavelength cloud size (bigger ones tend to be nearer by) height above Galactic plane (clouds that appear higher would be nearer by) In the outer galaxy it is much harder to determine the distance to clouds, so the uncertainties are larger. Best method so far: association of clouds with with H I...