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V (r) r = -a (r)

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< v φ. ). Define the circular velocity at radius r in the galaxy as V(r). Acceleration of the star moving in a circular orbit must be provided by a net inward gravitational ...
Motion under gravity Motions of the stars and gas in the disk of a spiral galaxy are approximately circular (vR and vz << vf). Define the circular velocity at radius r in the galaxy as V(r). Acceleration of the star moving in a circular orbit must be provided by a net inward gravitational acceleration:

V 2 (r) = -ar (r) r To calculate ar(r), must in principle sum up gravitational force from bulge, disk and halo.



ASTR 3830: Spring 2004

For spherically symmetric mass distributions: • Gravitational force at radius r due to matter interior to that radius is the same as if all the mass were at the center. • Gravitational force due to matter outside is zero. Thus, if the mass enclosed within radius r is M(r), gravitational acceleration is:

GM(r) ar = r2

(minus sign reflecting that force is directed inward)

† ASTR 3830: Spring 2004

Bulge and halo components of the Galaxy are at least approximately spherically symmetric - assume for now that those dominate the potential. Self-gravity due to the disk itself is not spherically symmetric… If you are familiar with vector calculus, Sparke & Gallagher 3.1 derives Poisson’s equation needed to calculate force from an arbitrary mass distribution. Note: no simple form for the force from disks with realistic surface density profiles…

ASTR 3830: Spring 2004

Rotation curves of simple systems 1. Point mass M:

GM V (r) = r

Applications: • Close to the central black hole†(r < 0.1 pc) • `Sufficiently far out’ that r encloses all the Galaxy’s mass eg image of the Galactic center Note: non-circular orbits and presence of massive stars Movie: Andrea Ghez’s group

ASTR 3830: Spring 2004

2. Uniform sphere: If the density r is constant, then: 4 3 pr r 3 4 pGr V (r) = r 3 M(r) =

Rotation curve rises linearly with radius, period of the orbit 2pr / V(r) is a constant independent of radius. † Roughly appropriate for central regions of spiral galaxies.

ASTR 3830: Spring 2004

3. Power law density profile: If the density falls off as a power law:

Ê r ˆ-a r(r) = r 0 Á ˜ Ë r0 ¯ …with a < 3 a constant, then: †

4 pGr 0 r0a 1-a 2 V (r) = r 3-a

For many galaxies, circular speed curves are approximately flat (V(r) = constant). Suggests that mass † density in these galaxies may be proportional to r-2.

ASTR 3830: Spring 2004

4. Simple model for a galaxy with a core: Spherical density distribution:

V H2 4pGr (r) = 2 r + a 2H • Density tends to constant at small r • Density tends to r-2 at large r

† Corresponding circular velocity curve is: V (r) = V H

Ê r ˆ aH 1arctanÁ ˜ r Ë aH ¯

† ASTR 3830: Spring 2004

Resulting rotation curve

ASTR 3830: Spring 2004

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