1. Celestial Mechanics PHYS390 (Astrophysics) Professor Lee Carkner Lecture 5

2. Questions 1) Can we achieve enormous resolving power by dramatically increasing N? Answer: No Explain: The more slits, the larger the grating has to be. Large gratings require the light to be more spread out, need higher intensity. 2) Can we achieve enormous resolving power by using very high orders? Answer: No Explain: Increase n, decrease brightness, need higher intensity to use higher orders.

3. Test #1  Next Wednesday, Dec 1  Multiple choice (25%)  Like in-class questions  Problems (75%)  Like PALs and homework  Sample equation sheet on web page  Just need pencil and calculator

4. Kepler’s Laws  In the 1600’s Johannes Kepler used Tycho Brahe’s data to find the laws of planetary motion  Kepler’s First Law   Half of the longest (major) axis is the semi-major axis called a   Half the distance between the foci is called c  e = c/a  e = 0 is circle (a=r)  e = 1 is line ca

5. Kepler’s Third Law  There is a relationship between the orbital size and period  Only in the solar system and if P is in years and a is in AU  P 2 =[4  2 /G(m 1 +m 2 )] a 3  Where m 1 and m 2 are the masses of the two objects

6. Reduced Mass  Consider two masses m 1 and m 2 located distances of r 1 and r 2 from the center of mass of the system   We can define the reduced mass,   = m 1 m 2 /(m 1 +m 2 )  r 1 = -(  /m 1 )r r 2 = (  /m 2 )r   Can think of as the reduced mass (  ) orbiting the total mass (M)

7. Angular Momentum  Orbiting objects have angular momentum  In general:  e.g., a single mass, m in a circular orbit of radius, r:  so for two bodies, total L is: L = m 1 r 1 v 1 + m 2 r 2 v 2  L =  [GMa(1-e 2 )] ½

8. Velocity  Energy of reduced mass  The total system energy is then: E = -(Gm 1 m 2 /2a)  v 2 = GM[(2/r)-(1/a)]

9. Visual Binaries  Consider a binary star composed of two masses in two elliptical orbits around the center of mass (cm) m 1 /m 2 = a 2 /a 1   1 = a 1 /d  m 1 /m 2 =  2 /  1  even if we don’t know the distance, if we can measure  ’s we can find the mass ratio cm 

10. Finding Mass  If we do know the distance, we can get a 1 and a 2  P 2 = (4  2 a 3 )/[G(m 1 +m 2 )]  Have to use m 1 +m 2 and m 1 /m 2 to pull out m’s separately

11. Inclination   The observed subtended angle is now ά =  cos i  then a =  d = άd/cos i  m 1 +m 2 = (4  2 /G)(d/cos i) 3 (ά 3 /P 2 )

12. Next Time  Read: 7.2-7.3  Homework: 2.6, 2.14, 7.4, 7.7, 7.12,