Physics of AstronomyWinter Week 6 - Thus.16.Feb.2006

Astro-A: Universe 4 – Gravity and Orbits

Pre-lab – brief discussion

Astro-B: Finish C&O Ch.2

2:30 Seminar

4:00 Lab in CAL 1234

Let’s look at the sky together after lab!

Looking ahead (remember to take Universe Ch.4 online quiz)

Astro-A: Universe 4 – Gravity & Orbits

Ptolemy: Circular orbits about Earth

Copernicus: Circular orbits about Sun

Kepler: Elliptical orbits about Sun

We derived Kepler’s 3d law from Newton’s 2d law:

F=ma

Gravitational force acceleration in circular orbit

F=GmM/r2 a = v2/r

Solve for v2:

Speed v = distance/time = 2r/T. Plug this into v2 and solve for T2:

This is Kepler’s third law: T = period and r = orbit radius.

For objects orbiting the Sun, Kepler’s law simplifies to a3 = p2,

where a=radius in AU and p=period in years

A satellite is placed in a circular orbit around the Sun, orbiting the Sun once every 10 months. How far is the satellite from the Sun? 2

3 2 10 a = p = _______

12

a ______

Sidereal (real: P) and Synodic (apparent: S) periods:

A satellite is placed in a circular orbit around the Sun, orbiting the Sun once every 10 months. How often does the satellite pass between the Earth and the Sun?

1 1 1

'

1 1 1

1 1 110 1

121

________________

________________

sidereal period Earth s sidereal year synodic period

P E S

S

SS

We can use Newton’s gravity to approximate the size of a black hole!

2

*

* ________grav

Energy Force distance

GmME r

r

Knowing the gravitational force between two bodies m and M, we can find their gravitational energy:

In order for an object (say, m) to escape M’s gravity,

It needs sufficient kinetic energy K=1/2 mv2 …

Use energy conservation to find the size of R of a black hole:

21

2____________

Gravitational energy kinetic energy

GmMmv

rSolve for r

Not even light can escape (v=c) if it is closer than r to a black hole. This is the Schwarzschild radius:

R(for v=c) =_____________________

Lab

height and speed vs time

0.00

5.00

10.00

15.00

20.00

25.00

0 0.5 1 1.5 2 2.5

time(s)

height(m)

speed(m/s)

Hand in before lab:• Excel practice results• Pre-lab

Lab this afternoon• Read lab guidelines• Take notes in a bound

notebook• Write up report this

weekend

Kinetic and Potential energy vs time

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5

time(s)

Kinetic(J)

Potential(m/s)

Astro-B: C&O Ch.2: Gravity & orbits

2 2 34GMT R

Kepler’s laws (for M>>m) Generalization (for M~m)

K1: orbits are elliptical - about center of mass

K2: equal areas in equal times - conservation of L

K3:

Center of mass

Virial Theorem

2 2 34G M m P a

Center of Mass reference frame

Total mass = M = m1+ m2

Reduced mass =

Total angular momentum L=r v = rp vp

1 2

1 2

mm

m m

21L GMa e (Pick any point, e.g. perihelion: rp and vp) p.50

Virial Theorem <E> = <U>/2

where <f> = average value of f over one period

For gravitationally bound systems in equilibrium, the total energy is always one-half of the potential energy.

Ex: Use K3 in Eorbit = T + U for circular orbit:

New HW: 2.8, 2.9

Looking ahead

Monday Tuesday Wed Thus Fri