In-Class Worksheet on the Galaxy

Ay 20 / Fall Term 2014-2015 Hillenbrand In-Class Worksheet on The Galaxy

Today is another collaborative learning day. As earlier in the term when working on blackbody radiation, divide yourselves into groups of 2-3 people. Then ﬁnd a broad space at the white boards around the room. Work through the logic of the following two topics.

Oort Constants. From the vantage point of the Sun within the plane of the Galaxy, we have managed to infer its structure by observing line-of-sight velocities as a function of galactic longitude. Let’s see why this works by considering basic geometry and observables.

• Begin by drawing a picture. Consider the Sun at a distance Ro from the center of the Galaxy (a.k.a. the GC) on a circular orbit with velocity θ(Ro) = θo. Draw several concentric circles interior to the Sun’s orbit that represent the circular orbits of other stars or gas. Now consider some line of sight at a longitude l from the direction of the GC, that intersects one of your concentric circles at only one point. The circular velocity of an object on that (also circular) orbit at galactocentric distance R is θ(R); the direction of θ(R) forms an angle α with respect to the line of sight. You should also label the components of θ(R): they are vr along the line-of-sight, and vt tangential to the line-of-sight. If you need some assistance, Figure 18.14 of C/O shows the desired result. What we are aiming to do is derive formulae for vr and vt, which in principle are measureable, as a function of the distance d from us, the galactic longitude l, and some constants. We are considering motions in the plane of the Galaxy only, so the orthogonal coordinate of galactic latitude b does not matter. (Remember l and b from the ﬁrst week of class?)

• Look at your ﬁgure and write the simple expression for vr in terms of θ, θo, (all usually expressed in km/s) and sines and/or cosines of α and l. You can make the substitution θ = Ω × R and also use “the sine theorem” (somebody in your group will remember it) to get rid of the α. You should wind up with vr = (Ω − Ωo) Ro sinl.

• Now write the simple expression for vt also in terms of θ, θo, and sines and/or cosines of α and l; then, as above, substitute in the expression involving Ω’s and R’s for the θ’s. For the triangle connecting: 1) the GC, 2) the perpendicular from the GC to vr, and 3) the Sun, you can deﬁne cos l in terms of d, R sinα, and Ro. Making this substitution to again get rid of the α, you should wind up with vt = (Ω − Ωo) Ro cosl − Ω d.

• Now, what we are actually seeing in any observed values of vr and vt are the projec- tions of the velocity vector difference between θ(Ro), representing the Sun’s orbital motion, and θ(R) representing the object’s motion, i.e. there is a vector difference ∆θ = θ(Ro) − θ(R). We can consider what happens at different distances from us, and along different lines of sight or longitudes relative to the GC. For example, at d << Ro or close to the Sun, we can consider the case when R ≈ Ro This lets you deal with the (Ω(R)−Ω(Ro)) term in the two velocity expressions above by considering when (R − Ro) is small and employing a Taylor expansion to substitute. Work this out to get an expression for (Ω(R) − Ω(Ro)) that involves evaluating dΩ/dR and produces

two terms involving θ’s and/or R’s and/or dθ/dR|R=Ro each multiplied by a factor of (R − Ro).

• Next go back to your figure and find a simple trigonometric substitution for the (R−Ro) factor, valid when R ≈ Ro. This should involve d and l only. • Now for the crux of things. The Oort constants A and B are defined as follows

A = 1/2[ θo − dθ | ] – representing the shear Ro dR R=Ro B = −1/2[ θo + dθ | ] – representing the vorticity Ro dR R=Ro

Note that B = A − Ωo. What would be useful astronomical units?

• Now consider from above that vr = (Ω − Ωo) Ro sinl and plug in what you derived for (Ω(R) − Ω(Ro)). Now look at the Oort constants and ﬁnd a relation for vr in terms of A, d, cosl, and sinl.

• Similarly, consider from above that vt = (Ω − Ωo) Ro cosl − Ω d. For the ﬁrst term you can jump directly to the same A, d, and cosl expression that resulted from considering vr. For the second term you can consider that Ω ≈ Ωo.

• After minor trig manipulation (think half angles) of vr and similar for vt but remem- bering that (A − Ωo) = B, you should be able to make your expressions look like:

vr = A d sin(2l) and

vt = A d cos(2l) + B d A further point for vt is that what we actually observe is proper motion or µ = vt/d, so as usual you need to know the distance in order to get the velocity. • Here is how what you have derived is relevant. The expression for the two components of the velocities (radial and tangential) are simple waves – the double sine and double cosine. So you can imagine taking observations around the galaxy at a range of l values and plotting vr vs l, which has amplitude A, and also µ vs l, which has amplitude A and mean value B. Draw these plots just to make sure you understand the concept. From observations involving lots of data along diﬀerent lines of sight, with substantial scatter, it is determined that:

A = 15 km/sec/kpc B = −10 km/sec/kpc Is there a conclusion to be drawn from the fact that a double sine wave is what is actually observed? What complications of real galactic dynamics make this nice logic not exactly correct? • Finally, we can look at the sum and diﬀerence of A and B. – What is A − B both symbolically (look at the deﬁnitions) and numerically (look at the numbers given above)? Why measure the proper motion of the GC? – What is A + B both symbolically and numerically? What does sign imply?

Rotation Curves and Spiral Structure. A classic issue in astronomy is the difference between the expectations and the actual observations of rotation (the θ’s above) as a function of galactocentric radius (R’s above) – for our Galaxy as well as in other galaxies. We will consider solid body rotation and differential (Keplerian) rotation. • Under solid body rotation, how should speed vary as a function of galactocentric radius R? Draw a sketch. • Evaluate using their definitions the Oort constants in this scenario. What does the disagreement with the numerical values given above imply? • Under Keplerian rotation, how should speed vary as a function of galactocentric radius R? Add the curve to your sketch. • Evaluate the Oort constants in this scenario. You should have something that agrees in sign with the numerical values above, and could agree exactly. What is θo(Ro) if Ro = 8 kpc? If the actual value of θo(Ro) = 220 km/s, what is implied? • (skip these next two bullets if time is running short and move on to the punchline) Getting more realistic, you can consider the expected rotation curve for a disk of finite thickness 2h and constant density ρo. For example, the disk thickness 2h can be set to 1/10Rgal for a “thin disk”. If you normalize in radius R to the disk radius, Rgal, you can consider a curve out to 10Rgal. Assuming circular orbits and rotational support of the disk, sketch the expectation for θ vs R/Rgal. • For the Milky Way, we can let the disk radius be 12 kpc and the mass surface density 2 be 75 M /pc . Re-label or re-draw your sketch with some actual numbers. • The reality might be that the observed rotation curve is flat (i.e. dv/dR = 0) beyond the intersection of the rising solid body curve and the declining Keplerian curve – all the way out to 100 kpc. Evaluate the Oort constants in this scenario. What is θo(Ro) if Ro = 8 kpc? If the actual value of θo(Ro) = 220 km/s, what is implied? How much mass would be required to produce this kind of curve? Since we don’t see this missing mass, it is called “dark matter”.

P.S. Anything not ﬁnished in Class on Wednesday should be completed outside of class. This is not homework, though you should understand the end results in each case, especially if you are headed to Ay21 on galaxies and cosmology.