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4 Pages

### Oort_diagram

Course: ASTR 553, Fall 2008
School: UVA
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Word Count: 616

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Galactic Circular Rotation. Derivation of Oorts constants: A &amp; B S l r V(Ro) V(R) l.o.s. V = V(Ro) V(R) R dV/dR r cos l V(Ro) Ro R V(R) l 90 l l.o.s. Fig 2 shows the velocity vectors of the sun and star (red) and their relative velocity (blue), which is what we witness. R = Ro R r cos (r sin l ) / l l.o.s. ve vr V V Radial velocity components projected onto the line of sight: v...

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Galactic Circular Rotation. Derivation of Oorts constants: A & B S l r V(Ro) V(R) l.o.s. V = V(Ro) V(R) R dV/dR r cos l V(Ro) Ro R V(R) l 90 l l.o.s. Fig 2 shows the velocity vectors of the sun and star (red) and their relative velocity (blue), which is what we witness. R = Ro R r cos (r sin l ) / l l.o.s. ve vr V V Radial velocity components projected onto the line of sight: v r = qVo cos l - DV sin l GC v t = - qVo sin l - DV cos l Fig 1 shows the global situation: a circular rotation velocity field of amplitude V(R) about the GC with the sun at S. Consider a star at galactic longitude l, distance r from the sun: it . subtends (r sin l) / Ro at the GC, and its velocity also subtends w.r.t. the solar velocity. It is closer to the GC by R r cos l and differs in V by R dV/dR. +ve vr +ve vt ve vt Fig 3 zooms into the difference velocity (blue). We want to find its projection parallel (vr: doppler) and transverse (vt : proper motion) to the line of sight (l.o.s.). To do this, make use of its (green) components parallel and perpendicular to the solar velocity (amplitudes V and Vo). v r = qVo cos l - DV sin l = Vo dV r sin l cos l r sin l cos l Ro dR V dV = r sin l cos l o = A r sin 2l R o dR V dV where A = 1 o is Oort' s first constant. (sheer) 2 R R o dR o Transverse velocity components projected onto the sky (proper motion): Vo dV r sin 2 l r cos2 l Ro dR 1 2 v t = - qVo sin l - DV cos l = - dV V = - 1 + o (cos2 l + sin 2 l) r 2 dR R o dV V = - 1 + o + r 2 dR R o 1 2 dV Vo 2 2 r (cos l - sin l) dR R o o dV V r cos 2 l = B r + A r cos 2 l R o dR V dV where B = - 1 o + is Oort' s second (rotation) constant. 2 R R o dR o A measures the local sheer: the degree to which stars slide past each other. E.g. solid body has no sheer, since dV/dR = V/R, so A = 0. Note a flat rotation curve (dV/dR = 0) does have sheer, with A = V/R. B measures local rotation, or vorticity. It comes from the curl of the velocity field: B = V. E.g. solid body is pure rotation...

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