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Lecture Six: the Milky Way: Kinematics

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Lecture Six: the Milky Way: Kinematics Physics of Galaxies 2014 Lecture Six: The Milky Way: Kinematics There will be a guest lecture on Tuesday 20th May (11-13) - Peter Barthel will give the lecture on AGN http://www.astro.rug.nl/~etolstoy/pog14 (his area of research interest) Sparke & Gallagher, chapter 2 6th May 2014 1 2 Milky Way Disc Rotation in a Disc Solid-Body 3 4 Differential Rotation Differential Rotation The Galaxy rotates differentially, which means that stars closer to the center take less time to complete their orbits around the Galaxy than those farther out If the Galaxy rotated as a solid (or rigid) body, then all stars in circular motion would take the same time to complete their orbits around the Galaxy Solid-Body 5 6 Stellar motions in the Milky Way disc: Stellar motions in the Milky Way disc: Galactic Rotation radial velocities To a good approximation stars and gas in the disc of the Milky Way move in nearly circular paths about the Galactic centre. We can calculate the radial velocity of a star (or a gas cloud), V0 cos(90 l) Stars closer to the Galactic centre complete their orbits in assuming that it follows a circular orbit. − less time than than those further out At radius R0 the Sun orbits with speed V0, while a star at P at DIFFERENTIAL ROTATION radius R has orbital speed V(R). The star moves away from us at speed V = V cos ↵ V sin l r 0 sin A sin B sin C This was first discovered from the proper motions of nearby − = = stars: we typically see stars orbiting in the same direction a b c but those closer to the Galactic centre pass us in their using the sine rule, we have sin l/R = sin(90 + ↵)/R0 R V cos ↵ orbits; those further away fall behind. cos ↵ = 0 sin l R V V0 and so V = R sin l This effect was already noticed in around 1900, but it wasn't until r 0 R − R 1927 that Jan Oort explained this as an effect of Galactic rotation. ✓ 0 ◆ If the Milky Way rotated rigidly, the distance between stars We can take advantage of this orderly motion to map out the distribution from measured velocities. would not change, and Vr would always be 0. In fact stars further from the centre take longer to complete their orbits; the angular speed V/R drops with radius R. 7 8 V V0 V/R decreases Vr = R0 sin l tells us R − R for larger R ✓ 0 ◆ o o Vr is positive for nearby For 180 < l < 270 Vr is objects where 0 < l < 90o always positive. o o For 90 < l < 180 Vr is pattern of first quadrant always negative (R>R0). repeated with sign reversed 9 10 Stellar motions in the Milky Way disc: Stellar motions in the Milky Way disc: radial velocities a2 = b2 + c2 2bc cos(A) proper motions − When the star or gas cloud P is very close to the Sun S, then The proper motion of a star at P relative to the Sun, S can be V0 cos l 2 d ≪ R and R ≈ R0 and we can neglect terms in d , using the calculated in a similar way. The tangential velocity is: cosine rule. 2 2 V0 cos(90 l) R = R 2R0d cos l V = V sin ↵ V cos l − 0 − t − 0 R0 cos l = R sin ↵ + d gives R R0 d cos l ⇡ − V V0 d Vt = R0 cos l V R − R0 − R V cos(90 ↵) V V0 then becomes, for small ✓ ◆ − Vr = R0 sin l R − R difference in V/R ✓ 0 ◆ Close to the Sun, R0 - R ≈ d cos l, and so Vt varies almost linearly with d(V/R) δ(V/R) δ(V/R) distance, d. = R0 sin l δ(V/R) = = dR δR R R0 R d(V/R) d 1 d(RV ) V cos ↵ − Vt d cos(2l) d(V/R) ⇡ − 2 dR − 2 R dR substitute R R = d cos l ✓ ◆ R0 R0 sin l (R R0) − 0 − ⇡ dR − remembering sin 2✓ =2sin✓ cos ✓ d [A cos(2l)+B] R d(V/R) ⌘ d sin(2l) dAsin(2l) B is the second of Oort's constants, and is measured to be ⇡ − 2 dR ⌘ ✓ ◆R0 -12.4±0.6 km/s/kpc A is one of Oort's constants, and is measured to be 14.8±0.8 km/s/kpc it measures the local vorticity, or angular momentum gradient in the disc. it measures local shear, or deviation from rigid rotation 11 12 Differential Rotation Stellar motions in the Milky Way disc: the Oort constants Combining A and B, we see that the magnitude of the circular velocity at R0 (~Sun) is: v = R (A B) c 0 − and the slope of the rotation curve at R0 is: dv c = (A + B) dR − R0 Note that the proper motion in the l direction is µ l = v t /d , so µl = B + A cos 2l A is the local (azimuthal) shear in the Milky Way’s rotation curve: the deviation from solid- body rotation B is the local vorticity, the gradient in the angular momentum in the MW disk locally — the tendency for stars to circulate around a point Recent results: A=14.8±0.8 km s-1 kpc-1 B=–12.4±0.6 km s-1 kpc-1 1 So the rotation curve appears to be gently falling and vc = 218(R0/8 kpc) km s− 13 14 Stellar motions in the Milky Way disc: Measuring rotation speeds in the the rotation curve outer Galaxy is harder. Need to find the parallax distances to If we can measure Vr for stars of known distance through out the disc of the Milky Way we can work backward to find V(R), the rotation curve of the Milky Way. associations of young stars. Their Vr is then measured from the Unfortunately, visible light from disc stars and clusters is blocked by dust. Radio waves can travel emission lines of gas around the through dust, and we receive emission in the 21 cm hyperfine transition of atomic hydrogen (HI) stars. The stellar distances are from almost everywhere in the Galaxy. But distances are not straight forward to determine. often not very well determined, but they are good enough to show For the inner galaxy (R < R0), use Tangent-Point method. The tangent point method fails that the rotation speed V(R) does V V not decline much in the outer The angular speed, V/R, drops with radius, so V = R sin l 0 for R ≤ R0, because this is gas r 0 R − R Galaxy and may even rise. ✓ 0 ◆ in the bar, which follows oval orbits. tells us that when we look out in the disc along a fixed direction with The gravitational pull of extra o 0 < l < 90 the radial speed Vr(l,R) is greatest at the tangent point (T). mass in spiral arms can easily This is where the line of sight passes closest to the Galactic centre. change the velocity of gas passing through them by 10-20km/s. If a tangent point R = R0 sin l V (R)=Vr + V0 sin l falls close to an arm, then the rotation speed found will different from the average so if there is emitting gas everywhere in the disc, we can find V(R) by speed of an orbit at that measuring the largest velocity seen in emission for each longitude, l. Galactocentric radius. 15 16 Milky Way Rotation Curve 17 18 2 2 2 NGP v = U + V + W Local stellar kinematics 391 | | G.C. The Galactic Disc in 3D 2 2 2 Non-Circular motions v = 10p +5.2 +7.2 =Local 13. stellar4 kinematics 391 U W Stars in the Milky Way disc do not move in perfectly circular orbits. We | | know this by carefully studying the solar motion, which is the motion of Rot. So we need to find the V motion under p V the Sun with respect to the Local Standard of Rest (LSR). the condition of no random motions: 1 The Local Standard of Rest (LSR) is defined as the frame rotating with the Galaxy that has V =5.2 0.6kms− ± zero deviation from the circular rotation at the radius of the Solar Neighborhood. So the magnitude of the Solar motion is In the LSR frame, the velocity components of a star are: (U, V, W )=(vR,vφ ΘLSR,vz) –1 Local stellar kinematics 391 − |v⊙|=13.4 km s , in towards the Galactic Center, up towards the North Galactic Pole, Looking at the motions of nearby disk and away from the plane. stars, using Hipparcos proper motions Figure 3. The components U, V and W of the solar motion with respect to stars with different colour B ¹ V. Also shown is the variation of the dispersion S with and measured radial velocities, we colour. should see properties similar to the Sun. Figure 3. The components U, V and W of the solar motion with respect to stars with different colour B ¹ V. Also shown is the variation of the dispersion S with colour. 2 The U and W components are The trend of V⊙ with S is called independent of color (spectral type): asymmetric drift, the tendency of the mean rotation velocity (of some 1 U = 10.0 0.4kms− population of stars) to lag behind that of ± 1 W =7.2 0.4kms− the LSR more and more with increasing ± random motion of the population Figure 3.
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