19. Galactic Rotation

1. 19. Galactic Rotation The Galactic co-ordinate system is defined such that the Galactic midplane is defined by main plane of 21cm emission. The zero-point is defined by the direction towards the Galactic centre (GC), which is assumed to be coincident with Sagittarius A*.

2. The Galaxy’s rotation is observed to be clockwise as viewed from the direction of the north Galactic pole (NGP). Galactic co-ordinates are Galactic longitude, l, measured in the direction of increasing right ascension from the direction of the GC, and Galactic latitude, b, measured northward (positive) or southward (negative) from the Galactic plane.

3. The velocity system for objects in the Galaxy is defined by: Θ = Rdθ/dt, the velocity in the direction of Galactic rotation Π = dR/dt, the velocity towards the Galactic anticentre Z = dz/dt, the velocity out of the Galactic plane.

4. The flatness of the Milky Way system, as evidenced for example by the narrow band of the Milky Way visible from Earth, suggests that the Galaxy has been influenced by general rotation about an axis perpendicular to the Galactic plane. The expected rotation of the Galaxy should be similar to what is found for any central force law (e.g. the solar system), namely differential rotation. That is, the angular velocity of rotation,  = v/r, should depend upon r, the distance from the centre of the Galaxy. In some galaxies and in the innermost regions of our own Galaxy, solid body rotation occurs; here  = constant = X, and v = Xr, i.e. v increases linearly with increasing r. For Keplerian motion, v ~ 1/r½ , i.e.  ~ 1/r3/2. Assume circular orbits about the centre of the Galaxy in the plane, and define  = the circular velocity at distance R from the centre of the Galaxy, 0 = the circular velocity at R0 = the Sun’s distance from the Galactic centre, and l = the Galactic longitude of an object of interest.

5. The co-ordinates used here are defined so that the velocities in the direction of the Sun’s motion, away from the Galactic centre, and towards the north Ggalactic pole are , , and Z, respectively. The observed radial velocity of the object at l relative to the local standard of rest (LSR = the reference frame centred on the Sun and orbiting the Galactic centre in the Galactic plane at the local circular velocity) is given by (see diagram): vR = Θ cos α – Θ0 cos (90°–l) = Θ cos α – Θ0 sin l . where Θ is the circular velocity at distance R from the Galactic centre and Θ0is the circular velocity at R0, the Sun’s distance from the Galactic centre.

6. By the Sine Law: So . Therefore, since Outside the Galactic plane the radial velocity becomes: .

7. The observed tangential velocity of the object relative to the LSR is given by: vT = Θ sin α – Θ0 cos l (where vT is positive in the direction of Galactic rotation). But R sin α = R0 cos l – d, where d is the distance to the object. So and These are the general equations of Galactic rotation.

8. If Ω decreases with increasing distance from the Galactic centre, then for any given value of l in the 1st (0° < l < 90°) and 4th (270° < l < 360°) quadrants, the maximum value of Ω occurs at the tangent point along the line-of-sight, i.e. at Rmin = R0 sin l. In that case, d = R0 cos l, so: Rmin = R0 cos (90° – l) = R0 sin l . vR(max) = Θ(Rmin) – Θ0 sin l .

9. Taylor series approximations to the general formulae can be made for relatively nearby objects, where d << R0, in which case: and . But And, for d << R0, R0 R≈ d cos l . So, for nearby objects in the Galactic plane vR becomes:

10. or vR = Ad sin 2l = Ad sin 2l cos2b, outside the plane, where: is Oort’s constant A.

11. For the tangential velocity: or vT = Ad cos 2l + Bd, where: is Oort’s constant B.

12. Therefore: The two constants A and B are referred to as Oort’s constants: Thus, Also,

14. Evidence for the Sb nature of the Galaxy: a rotational velocity near 250 km/s and an absolute magnitude near MB ~ 21.

15. Recall that, for a given line of sight within the solar circle, a maximum value for Ω is reached at the tangent point, where: d = R0 cos l, Rmin = R0 sin l . i.e. vR(max) = Θ(Rmin) – Θ0 sin l . For d << R0, we have: So: or: vR(max) = 2AR0(1 – sin l) sin l

16. The actual relationship is a series, with the second order term generally being ~10% the magnitude of the first order term: e.g. Studies of the radial velocities of stars in the first and fourth quadrants in order to determine vR(max) generally yield values of AR0 lying in the range 135–150 km/s. Optimum Values for the Galactic Rotation Constants. A summary studies of the Galactic rotation constants to 1986 is given by Kerr & Lynden-Bell (MNRAS, 221, 1023, 1986). Despite the paper’s title (“Review of Galactic Constants”), the various estimates are summarized, not reviewed. Here we try to review the various estimates with a view to obtaining the optimum current values.

17. Oort’s A & B. The predicted effects of Galactic rotation on radial velocities and proper motions of nearby stars appear as a double sine wave dependence of the radial velocities with Galactic longitude and a double cosine wave dependence of the proper motions with Galactic longitude, the latter offset from zero by the Oort B term. The proper motion relationship is a modified version of the vT relation:

19. Both effects are clearly seen in the available radial velocity and proper motion data, but different studies have obtained values for A ranging from 11.6 km/s/kpc to 20.0 km/s/kpc, and values for B ranging from –7.0 km/s/kpc to –18 km/s/kpc. A proper motion study from the Lick Northern Proper Motion Program (Hanson AJ, 94, 409, 1987) yielded estimates of A = +11.31 ±1.06 km/s/kpc and B = –13.91 ±0.92 km/s/kpc, while a study by Schwan (A&A, 198, 116, 1988) using FK5 system proper motions yielded estimates of A = +12.32 (or 14.22) km/s/kpc and B = –11.85 km/s/kpc (no quoted uncertainties). Radial velocity studies tend to give larger estimates for A, but that is possibly because they sample a larger region of space where the approximations leading to Oort’s equations break down. Best estimates for A and B based only upon recent proper motion work are: A ≈ +12.5 ±1.0 km/s/kpc , and B ≈ –12.5 ±1.0 km/s/kpc .

23. The result has implications for the nature of the local Galactic rotation. For the case of solid-body rotation with Ω = constant, one predicts that: Neither prediction satisfies the observations, which means that differential rotation is confirmed. Alternatively, it is possible that Θ is constant, at least locally. In that case:

24. Currently available data do indicate that A ≈ –B, so a constant circular velocity does appear to exist locally. The case for (R) = constant = 0 is referred to as a flat rotation curve, and seems to be appropriate for nearly all spiral galaxies, not just the Milky Way. Available data from radio studies indicate that the rotation curve of our Galaxy is fairly simple. It seems to obey solid body rotation close to the Galactic centre, and turns into a flat rotation curve in the outermost regions, including the region at the Sun’s distance from the Galactic centre. Dynamical theory suggests that A and B should also be related through the parameters of the velocity ellipsoid, namely that:

25. for a flat rotation curve. As noted by Kerr & Lynden-Bell, many studies of the ratio give values for (σΘ/σΠ)2very close to 0.5, with typical values ranging from 0.36 to 0.50, and with late-type giants (representing a dynamically relaxed system) giving values of 0.49 to 0.50, closest to the result predicted for a flat rotation curve. Θ0. A value of Θ0 = 245 ±10 km/s results from the study of plunging disk stars and velocities of Local Group members. Past studies gave values from 184 km/s to 275 km/s. The value 250 km/s adopted by the IAU in 1964 was adjusted to 222.2 km/s by Kerr & Lynden-Bell, but such a small value cannot be reconciled with velocities of nearby galaxies nor with the data for high-velocity stars.

26. R0. Estimates for R0 listed by Kerr & Lynden-Bell range from 6.8 kpc to 10.5 kpc. Direct methods may not be capable of yielding a reliable value, given current uncertainties in the distance scale, so it is important to examine what indirect methods yield. A value for R0 can be derived using Oort’s constants A and B with Θ0, since: for the values cited earlier. If one is willing to accept values of A = –B = 14.0 ±1.0 km/s/kpc, the result is R0 = 8.75 ±0.72 kpc. One can also use the equation for maximum vR along the line of sight in the first and fourth quadrants:

29. It yields estimates of AR0 = 135 to 150 km/s, although Kerr & Lynden-Bell list values ranging from 103 to 156 km/s. Most estimates are susceptible to the adopted LSR velocity, and may contain systematic errors. Recent studies (those since 1974) yield values of AR0 = 118 ±15 km/s, corresponding to R0 = 9.44 ±1.42 kpc. Another method of obtaining R0 is through the use of zero-velocity stars, as illustrated in the diagram at right. Such objects have no net radial velocity with respect to the LSR, which means that they share the same Galactic orbital velocity as the Sun, i.e. Θ0. Their distances are given by: d = 2R0 cos l0

30. Thus: The method is susceptible to distance scale errors, to any local deviations from circular motion, to any errors in the adopted LSR velocity of the Sun, and even to slight errors in l0. Crampton et al. (MNRAS, 176, 683, 1976) used the technique to obtain a value of R0 = 8.4 ±1.0 kpc for B stars, not an unreasonable result. Interesting results have been derived using radio interferometry of the motion of H2O masers in the region of the Galactic centre (Reid et al. ApJ, 330, 809, 1988). A value of R0 = 7.1 ±1.5 kpc was obtained using Sgr B2(N), while a value of R0 = 10.8 ±4.8 kpc is quoted from the use of W51. Again the exact results are susceptible to the adopted LSR velocity of the Sun, as well as to the particulars of the model.

31. Estimates based upon the detection of planetary nebulae or Mira variables in the nuclear bulge may hold more credence. A value of R0 ≈ 8.1 kpc was obtained by Pottasch (A&A, 236, 231, 1990) using the planetary nebula luminosity function, although it is not clear how susceptible the value is to reddening corrections or to bias resulting from being centred on nearby bulge objects rather than the Galactic centre. Whitelock et al. (MNRAS, 248, 276, 1991) obtained a value of R0 = 8.6 ±0.5 kpc using Mira variables, noting, however, that the uncertainty in their result might be larger than the quoted value. Racine & Harris (AJ, 98, 1609, 1989) obtained R0 = 7.5 ±0.9 kpc using globular clusters in the nuclear bulge. It appears that recent estimates of R0 fall in the range from 7.1 to 8.6 kpc. All are susceptible to various problems. Ideally, however, any adopted value must be consistent with the values obtained for Oort’s constants, as emphasized by Kerr & Lynden-Bell.

32. Reid and Brunthaler (2004, ApJ, 616, 872) have measured the proper motion of Sgr A*, the radio source apparently associated with the Galactic centre, using the Very Long Baseline Array in New Mexico. They find two components, one in the Galactic plane and the other perpendicular to the plane, with values of μl = –6.379 ±0.026 mas yr–1 and μb = –0.202 ±0.019 mas yr–1, respectively. If one adopts values for the Sun’s motion relative to the Galactic centre of 250.3 ±8.6 km/s in the Galactic plane and +7.0 ±0.2 km/s perpendicular to the plane, the resulting best estimates are R0 = 8.3 ±0.3 kpc from μl and R0 = 7.3 ±0.9 kpc from μb, both consistent with a value of R0 = 8.0 ±0.5 kpc. Parameter IAU (1964) IAU (1985) Current? R0 10 kpc 8.54 kpc 8.5 kpc A +15 km/s/kpc +14.45 km/s/kpc +12.5 km/s/kpc B –10 km/s/kpc –12.0 km/s/kpc –12.5 km/s/kpc 0 250 km/s 222.2 km/s 245 km/s

33. Expectations from the equations of motion for Galactic rotation. In the 1st quadrant (0° < l < 90°) stars recede from the Sun, in the 2ndquadrant (90° < l < 180°) they approach from the Sun, in the 3rdquadrant (180° < l < 270°) they recede from the Sun, and in the 4th quadrant (270° < l < 360°) they approach the Sun.

34. A typical orbit for a star in the Galaxy can be pictured as epicyclic motion of frequency κ superposed on circular motion of frequency Ω. When κ = 2Ω the orbit is an ellipse. Since κ(R)≠ 2Ω(R) in most cases, the orbits are roseate, something like what is produced by a spirograph. Cyclical motion perpendicular to the Galactic plane also occurs.

35. The random motion of nearby stars relative to each other produces the observed velocity dispersions for various stellar groups. Stars in the Galactic bulge appear to exhibit no preferred direction or orbital inclination, so define a spheroidal distribution.

36. 20. Galactic Structure Studies 21–cm Radiation of Hydrogen. 21-cm emission from neutral hydrogen gas is used to locate hydrogen clouds in the Galactic plane using information on their Doppler shifts in conjunction with the relationship for the LSR-corrected radial velocity of an object in an orbit about the Galactic centre, i.e. It is reasonable to assume that Ω(R) decreases with increasing R, so that the maximum radial motions of hydrogen clouds along the line of sight for 0° < l < 90° (1st quadrant), and the minimum radial motions of hydrogen clouds along the line of sight for 270° < l < 360° (4th quadrant) occur at Rmin = R0 sin l, i.e. the tangent point, where:

37. The maximum and/or minimum observed radial velocities of hydrogen clouds along such lines of sight in the 1st and 4th quadrants, respectively, must originate from gas located at the tangent points, if there is any. In general terms there will always be some hydrogen gas at the tangent points, but their maximum and/or minimum radial motions will be the vector sum of their orbital motion about the galactic centre and their random space motions, which typically average 15–20 km/s. If Θ0 is known, one can construct a relationship for Θ(R0 sin l) = vR(max) + Θ0 sin l for 270° < l < 90°. The relationship can be extrapolated to R > R0 using mainly theoretical expectations (see Blitz, ApJ, 231, L115, 1979), and the resulting Θ(R)relationship can then be used with 21-cm observations of neutral hydrogen cloud velocity peaks to determine the R-distribution of the higher density regions of hydrogen gas in the Galactic plane. The velocities must first be adjusted to the LSR, which is probably the most uncertain step.

38. The method works reasonably well in the 1st and 4th quadrants, where the Θ(R)relationship is well established, and also gives fairly consistent results for the 2nd and 3rd quadrants. For any line of sight there are always ambiguities in distance for clouds of a specific radial velocity, since simple mathematical analysis indicates that clouds at two different distances symmetric about Rmin must have identical radial motions. Such ambiguities are resolved by mapping the clouds in Galactic latitude b, since nearby clouds should subtend a larger angular extent than distant clouds. The method also breaks down at l = 0° (towards the Galactic centre) and l = 180° (the anticentre), since there are no predicted radial motions of material arising from Galactic rotation in those directions. Within such constraints, however, 21-cm maps exhibit a distinct spiral arm picture that is in fairly good agreement with the maps of other spiral arm indicators, at least in the solar neighbourhood.

39. Evidence for a warp in the distribution of neutral hydrogen is also fairly obvious, with the Galactic plane being warped north of b = 0° towards l ≈ 90° and south of b = 0° towards l = 270°, by perhaps 0.8 kpc at 1.5 R0. It is usually explained as the result of an interaction of the Galaxy with the Large Magellanic Cloud, the Small Magellanic Cloud, and the other objects in the Magellanic Stream. The feature should not be confused with Gould’s Belt, which appears as a local tilting of the nearby spiral arm — below the Galactic plane in the direction of the anticentre and above the plane towards the Galactic centre. It is also noteworthy that the hydrogen abundance decreases beyond the solar circle, a feature also detected in radio continuum data and seen in the decreased numbers of H II regions and massive stars outside of the solar circle.

41. Once the solar LSR velocity is established, it is possible to establish likely distances to Galactic objects from their radial velocities, i.e. using the equations of Galactic motion. But all such previous efforts have used an incorrect solar LSR velocity.

48. Molecular Lines. Radiation from the ubiquitous CO molecule can also be used to trace spiral features, although most of the early studies concentrated upon the northern hemisphere. CO originates mainly in large molecular clouds, as does most neutral hydrogen, so different maps from CO and 21-cm radiation should complement each other. Radio molecular radiation is also generated at the frequencies of OH and H2O molecules in regions of star formation, so studies at such frequencies usually yield different information about Galactic structure than do 21-cm or CO maps.