Physics of Astronomy, week 4, winter 2004 Astrophysics Ch.2 Star Date Ch.2.1: Ellipses (Matt #2.1, Zita #2.2) Ch.2.2: Shell Theorem Ch.2.3: Angular momentum (J+J, #2.7) #2.11: Halley’s comet Learning plan for week 5

Ch.2.1: Ellipses (Matt #2.1, Zita #2.2) Make an ellipse: length of string between two foci is always r’ + r = 2a. Eccentricity e = fraction of a from center to focus.

#2.1: Derive the equation for an ellipse. Distance from each focus to any point P on ellipse: r 2 =y 2 +(x-ae) 2 r’ 2 =y 2 +(x+ae) 2 Combine with r+r’=2a and b 2 = a 2 (1-e 2 ) to get

#2.2: Find the area of an ellipse. so y goes between and x goes from (-a to +a) Area =

Ch.2.2: Shell Theorem (p.36-38) The force exerted by a spherically symmetric shell acts as if its mass were located entirely at its center. The force exerted by the ring of mass dM ring on the point mass m is Where s cos  = r - R cos  and s 2 = (r - R cos  ) 2 + (R sin  ) 2 and dM ring =  (R) dV ring and dV ring = 2  R sin  R d  dR

Substitute this into dF and integrate Change the variable to u = s 2 = r 2 + R 2 - 2rR cos  Solve for cos  sin  Substitute these in and integrate over du to get

Density = mass of shell / volume of shell  (R) = dM shell / dV shell So dM shell =  (R) dV shell = 4  R 2  (R) dR Which is the integrand of So the force on m due to a spherically symmetric mass shell of dM shell : The shell acts gravitationally as if its mass were located entirely at its center.. Finally, integrating over the mass shells, we find that the force exerted on m by an extended, spherically symmetric mass distribution is F = GmM/r 2

Force and Angular momentum Force = dp/dt = m dv/dt + v dm/dt F = ma + v dm/dt If  F = 0, then dp/dt=0 and p=constant: Momentum conservation Torque = dL/dt = d(r x p)/dt = (dr/dt) x p + r x dp/dt = v x p + r x F If v || p and r || F, then  torque = 0 = dL/dt L = constant: Angular momentum conservation for any central force

Center of Mass reference frame Total mass = M = m 1 + m 2 Reduced mass =  Total angular momentum L=  r v =  r p v p

Virial Theorem = /2 where = average value of f over one period Example: For gravitationally bound systems in equilibrium, the total energy is always one-half of the potential energy.

Mon.2.Feb: Introduction to Astrophysics Ch.3 Universe Ch.5.1-3, #6, 11 (Jared + Tristen) Universe Ch.5.4-5, #25 (Brian + Jenni) Universe Ch.5.6-8, #27, 29 (Erin + Joey) Universe Ch.5.9, #34, 36 (Matt + Chelsea) Universe Ch.19.1, #25 (Annie + Mary) Tues.3.Feb: HW due on Physics Ch.6 Universe Ch , #34, 35 (Jared + Tristen) Universe Ch , Spectra -> T,Z, #43 (Erin + Joey) Universe Ch.19.6, L(R,T), #46, 50 (Annie + Mary) Universe Ch.19.7,8, HR, #52 (Brian + Jenni) Thus.5.Feb: HW due on Astrophysics (CO) Ch.2 CO 3.1, Parallax, #3.1 (Jared + Tristen) CO 3.2, Magnitude, #3.8 (a-d) (Erin + Joey) CO 3.3 Wave nature of light, #3.6 (Matt + Chelsea) CO 3.4, Radiation, #3.8 (e-g) (Brian + Jenni) CO 3.6, Color index, #3.13 (Annie + Mary) Learning Plan for week 5 (HW due Mon.9.Feb):