1. Oct. 4, 2007 1 Celebration of Orbits and how to understand/Use them What’s special about today? –We now “understand” orbits… and some of Newton’s genius… –We learned how to measure stellar masses (only ~1/4 into A-36) 50 th anniversary of Sputnik, the 1 st manmade object (satellite) put into orbit! –It’s the 50 th anniversary of Sputnik, the 1 st manmade object (satellite) put into orbit! (Oct. 4, 1957)Oct. 4, 1957 Derivation (as promised) of centripetal acceleration, a = V 2 /R Practice measuring masses of moons, stars, and galaxies Introducing the most extreme orbit: around a black hole… when the orbital speed V = c, the speed of light… More on uncertainties and how to report them… (not in class…)

2. Oct. 4, 2007 2 How to derive centripetal acceleration? Recall (last time) that centripetal acceleration of mass m in circular orbit about mass M at velocity V is a = V 2 /R which gives F = GMm/R 2 = m a = V 2 /R and thus V 2 = GM/R. But Why is a = V 2 /R? M R m V V’ ΔΘ Mass m moves with velocity V to V’ by changing direction by angle ΔΘ. Thus change in velocity is ΔV over time Δt during which m moves over path length p. But since angle is small, it is approximately ΔΘ = p/R = V Δt/R since p = V Δt. But because also the triangle of V changing to V’ shows that ΔΘ = ΔV/V, and since acceleration a = ΔV/Δt, we have a = V ΔΘ/Δt = V(V Δt/R)/ Δt or, --- voila --- a = V 2 /R p V V’ ΔΘΔΘ ΔVΔV a = ΔV/Δt Note: This derivation is optional; but you can now understand both why Orbits “work” – where the Centripetal force = m a comes from – and how Newton invented Calculus!

3. Oct. 4, 2007 3 So, again, Newton’s form of Kepler’s 3 rd Law But orbital velocity V around an orbit with “radius” or semi-major axis R is just V = 2π R/P = circumference/period So substituting this for V we have P 2 = (4π 2 /GM) R 3, with G = 6.8 x 10 -11 for M(in kg), R( in meters) Or, much more simply (and useful), P 2 = R 3 /M which must be used for P(in yrs), R(in AU), M(in solar masses)

4. Oct. 4, 2007 4 Using orbits to work out some Masses & other things… Blackboard exercises to derive M = R 3 /P 2 that we do together as well as other neat things (practice quiz questions…): –Mass of Earth from what we learned in school about Moon… Period of Sputnik’s orbit given Earth radius (book, Appendix 2) and altitude of Sputnik above Earth (about 300km) Velocity the Russian rocket needed to put Sputnik in orbit –Mass of Sun from our (hypothetical) measurement of sidereal period of Mars (from its synodic period of retro motion) as P = 1.9 ~2years and our (hypothetical) parallax measurement of Mars orbit’s R = 1.5AU –Mass of our entire Milky Way Galaxy (“interior” to Sun) from someone telling us (for now…) our Sun is 8.5kpc from galactic center and that our Sun (and solar system) has a “galactic year” of 250 million years

5. Oct. 4, 2007 5 And what about the “ultimate” orbit: around a BH? Consider orbital velocity, V ES = 2πR/P for Earth orbiting Sun. With R ES = 1AU = 1.5 x 10 8 km and P = 1yr = 3 x 10 7 sec, Earth moves at V ES = 30km/sec in orbit about a M = 1 solar mass star (Sun) [work this out yourself…] But recall V 2 = GM/R so V = sqrt(GM/R), so V = X sqrt(M/R), where X = sqrt(G) which is just a constant number So we can ask what is M/R if orbital velocity is speed of light, or V = c = 3 x 10 5 km/sec, as for black hole(!) Use ratios: V BH /V ES = 3 x 10 5 /30 = 10 4 = sqrt[(M/R) BH /(M/R) ES ] so a BH has M/R value larger than for Earth-Sun by a factor 10 8 or (M/R) BH = 10 8 (M/R) ES Therefore, for the same M = 1 solar mass, a BH must have a minimum orbital radius or “size” R BH = 10 -8 R ES = 1.5km ! So… compress Sun to R =1.5km and it will “wink out” of sight as a Black Hole!

6. Oct. 4, 2007 6 Discussion of Measurement uncertainties (cont.) Measurements always have uncertainties, which can be estimated in our labs (and in your everyday “observations” of the world, economy, etc.) If make N measurements, m i, first derive their average or mean value A m = [m 1 + m 2 + … m N ]/N and then the deviation, D i, of each (i = 1…N)) measurement about the average, D i = m i – A m. Make a Table of m i and D i and then calculate the average of all the D i ‘s Report this average scatter, A D = [D 1 + D 2 + … D N ]/N as your 1 st estimate of overall uncertainty, “mean error” (M.E.) “Root Mean Square” (rms) error is even better estimate: rms = sqrt[ sum((D 1 ) 2 + ((D 2 ) 2 + …(D N ) 2 )/N] Your fractional error F.E. = M.E./A m (or = rms/A m ) is overall summary of your uncertainty. If you have 2 measured values (e.g. d and r in DL1 solar pinhole measurements), just report the F.E. for the larger of the two as overall F.E.