Lagrange points and tides

Question 1

what’s a hill sphere

Answer 1

the range from a planet at which its gravity dominates the orbital motion of nearby objects

Question 2

what’s a Lagrange point?

Answer 2

positions at which a third body can co-rotate with two larger masses (orbit the same centre of mass with the same period)

Question 3

Lagrange period formula

Answer 3

T² = 4π²/G(M₁+M₂) * a³
M₁: mass of sun
M₂: mass of planet
a: distance between M₁ and M₂
T: orbital period of third body

Question 4

how many Lagrange points are there and where are they

Answer 4

5 total
L₁ : between planet and sun (very close to the planet)
L₂: opposite side of the planet to L₁ (still very close to the planet)
L₃: 180° further ahead in the orbit of the planet
L₄: 60° further ahead of the planet’s orbit (for M₁&raquo_space;M₂)
L₅: 60° further behind the planet’s orbit (for M₁&raquo_space;M₂)

Question 5

whats special about Lagrange points 4 and 5

Answer 5

they are stable equilibria. If a satellite in those orbits was perturbed then the Coriolis force would move them back

Since points 1, 2, and 3 are unstable equilibria, satellites must actively orbit these points.

Question 6

what causes tides

Answer 6

the gradient of the gravitational force across a body.

Question 7

whats the difference in gravitational force from the moon and sun and which has a larger effect on tides

Answer 7

sun is 178x greater gravitational force

moon is 2.2x greater effect on the tides

Question 8

what are the consequences on the tides of the moon orbiting the earth

Answer 8

the earth’s rotation slows due to the rotational energy dissipating as a result of the friction between the ocean and the earth’s body.

one of the bulges in the ocean (high tide) is ahead of the line joining the earth-moon. this leading bulge pulls on the moon, increasing its orbital energy and moving it outwards

Question 9

in terms of angular momentum, why does earths spin slow down

Answer 9

there’s a net torque on the ocean bulges from the moon. this torque causes the angular momentum of earth to decrease. To conserve angular momentum this leads to an increase in the orbital angular momentum of the earth-moon system

Question 10

what’s the Roche limit

Answer 10

the minimum orbital radius of a satellite before it would break apart.

Question 11

why is the tidal bulge no aligned with the moon

Answer 11

draw earth (2D) rotating anti-clockwise. draw the moon somewhere and the first tidal bulge is slightly further anti-clockwise than the line joining the earth and moon. the gap is called tidal lag. its caused by the ocean being dragged along by the earth’s rotation

Question 12

whats spring tide and neap tide

Answer 12

spring tide: tides with the largest range between high and low tide

neap tide: tides with the smallest range between high and low tide

Question 13

how do you approximate the distance from a planet to L₂

Answer 13

find dϕ/dx and put it =0 (=0 at the lagrange points)
ϕ = U/m is the effective potential per unit mass
a: distance between planet and sun
d: distance from planet to Lagrange point

U/m = -G[ M₁/(x-x₁) + M₂/(x₂-x) ] - ½x²ω²
where M₂ is the planet’s mass.

dϕ/dx = G[M₁/(x-x₁)² + M₂/(x-x₂)² ] - xω² = 0

since M₁&raquo_space; M₂: x₁ –> 0, x₂ –> a, x= a+d
G[M₁/(a+d)² + M₂/d² ] - (a+d)ω² = 0

sub in ω = 2π/T = (GM₁/a³)^½

M₁/(a+d)² + M₂/d² = (a+d)M₁/a³
M₁/a²(1+d/a)² + M₂/d² = (a+d)/a * M₁/a²
M₁/a² * (1 -2d/a) + M₂/d² = (1+d/a) * M₁/a²
M₂/d² = 3M₁* d/a³
M₂/3M₁ = d³/a³
d = a (M₂/3M₁)^1/3