4.3 Orbits and Universe

Question 1

What is Kepler 1st law?

Answer 1

Each planet moves in an ellipse with the Sun at one focus

Question 2

What is Kepler 2nd law?

Answer 2

The line joining a planet to the centre of the Sun sweeps out equal areas in equal times

Question 3

What is Kepler 3rd law?

Answer 3

T² (period of the planet’s motion squared) is proportional to r³ (where r is the mean distance from the planet to the star), when the orbit is close to circular.

Question 4

What’s the proof for Kepler’s 3rd law?

Answer 4

F = GMm/r²
F = mv²/r - (circular motion)
GMm/r² = mv²/r
GM/r = v²
v = d/t = 2πr/T (where T is period or year)
GM/r = 4π²r²/T²
T²= = 4π²r³/ GM
∴ T²= = (4π²/ GM)r³
or T²∝r³

Question 5

What is dark matter?

Answer 5

Matter which we can’t see, or detect by any sort of radiation but whose existence we infer from its gravitational effects.

Question 6

What are the 2 conditions for a binary system?

Answer 6

1) Each body has to have a mass large enough to effect the other’s motion detectably.
2) Both bodies need to orbit around the same point (called the system’s centre of mass)

Question 7

How do you find the centre of mass of a binary system?

Answer 7

You can find the radius of the planet’s orbit of one or both of the bodies to determine its position in space.

Question 8

How do you find the radius of an orbit of bodies in a binary system.

Answer 8

Let the masses of the planets be M₁ and M₂ and d be the distance between the planets (d = r₁ + r₂)
r₁ = (M₂/ M₁ + M₂) x d
r₂ = (M₁/ M₁ + M₂) x d

To check this you can use (w is angular velocity)
r₁M₁w² = r₂M₂w²
as the centripetal forces are equal for both planets

Question 9

What is Kepler’s 3rd law for binary systems?

Answer 9

Both bodies have a periodic time of:
T = 2π√d³ / G(M₁ + M₂)

Question 10

What is the speed of an orbiting body?

Answer 10

V = wr (w = 2π/T)
V = 2πr/T
T = period (for earth it’s 1 year)
r = Radius of orbit

or
v =√(GM / r)
G = Universal constant of gravitation
M = Mass of bigger body that this body is orbiting around

Question 11

What equation can help find the radial velocity of an object?

Answer 11

Δλ/λ = v/c
Δλ = Change of Doppler shifted wavelength
λ = Unshifted wavelength
R = Radial velocity
c = Speed of light

Equation works best when v &laquo_space;c (Much less than c) as

V is negative when it approaches us

Question 12

What is the radial velocity of an object?

Answer 12

This is the component of a star’s velocity along the line joining it and the observer on Earth.

Question 13

How was hobble’s constant found?

Answer 13

Measurements of the radial velocity of galaxies plotted against the distance to the galaxy from Earth gave a straight line.
The gradient and the gradient of this v, D line is Hubble’s constant.

V∝ D
v = HD
H = Hubble’s constant = 2.2 x 10^-18 s^-1

Question 14

How was the age of the universe found?

Answer 14

v = x/ t
v = H.D
∴ H.D = D/t
t = 1/H
t = age of universe

Question 15

What is the critical density of the universe?

Answer 15

The overall mean density of the universe at which the expansion will slow to zero rate but only at infinite size.

Question 16

What is proof for the critical density formula?

Answer 16

1) Mass of the universe is M = Vρ = (4/3)πr³ρ
2) Out edge of the universe (shell) is moving away at v =Ho x r (Ho = Hubble’s constant)
3) Will the shell have enough energy to leave the sphere of the universe? The escape velocity is the velocity needed for a body (or shell) of mass m to reach infinity with no spare KE,
Initial KE of shell = PE at infinity - Initial PE (at r)
½mv² = 0 - (-GMm/r)
v²r = 2GM
4) Sub in values of v and M : (Ho r)²r = 2G(4/3)πr³ρ
5) Cancel r³ : ρ = 3Ho²/8πG

Question 17

How does the Higgs boson relate the dark matter?

Answer 17

Scientists are looking at the decay of a Higgs Boson to see if there are any unexplained decay mechanisms that could suggest the presence of dark matter.

Question 18

What is gravitational lensing and how does it relate to dark matter?

Answer 18

Non-baryonic dark matter can be detected by using gravitational lensing which observes the distortion of light from a distant source that has been bent by the mass of the galaxy as per Einstein’s Theory of General Relativity.

The distortion isn’t consistent with amount of observed mass which suggests there is extra dark matter we can’t see.

Question 19

What are the 3 conditions for a mutual centre of mas (COM )

Answer 19

1) The COM must lie on the line joining the centres of the two bodies
2) The COM must lie between the two bodies as the centripetal force points towards it.
3) The angular velocity of each body must be equal.

Question 20

What are the conditions for the doppler effect equation?

Answer 20

Δλ/λ = v/c

Only works when v &laquo_space;c (Much less than c).
As V approaches c relativistic effects must be taken into account and the equation is no longer accurate.