Tidal Evolution of the Earth-Moon System

The goal of this project is to write a code to integrate the evolution of the Earth-Moon
system, from the present day configuration, back into the past. I suggest that you write the code
in python 3 in a juypter notebook, but you can use a different language if you prefer. Using the
equations and parameters given below, you should find that the Earth-Moon separation goes to
zero, i.e., that the two bodies collide, about a billion years into the past.
Ocean tides, experienced on a daily basis by beachgoers, are caused by the gravitational
attraction of the Moon, and to a lesser extent, the Sun. Lunar tides are known to increase both
the length of day and the length of the Lunar month; the Lunar torque −T$ produced by the
Moon’s gravity acting on the tidal bulge in the oceans and solid body of Earth reduces the spin
angular momentum S⊕ of the rotating Earth, and adds the same amount to the orbital angular
momentum L$ of the Moon, with no change in the sum LEM ≡ S⊕ + L$. The solar torque
T associated with the solar tide increases both the length of day and the length of the year; it
also reduces, very slightly, LEM , increasing the orbital angular momentum L⊕ of Earth by the
same amount:
dt = T , (1)
dt = −T − T$, (2)
dt = T$. (3)
You can use a simple tidal model, in which the the Lunar tidal torque is
T$ =

. (4)
In this expression, G = 6.67 × 10−8
is Newton’s gravitational constant, m$ =
7.349×1025 g is the Lunar mass, R⊕ = 6, 371 km is the radius of Earth, and a$ is the semimajor
axis of the Lunar orbit. The quantity k2 = 0.298 is the (dimensionless) Love number of Earth; it
encapsulates how rigid the Earth is. The dimensionless tidal quality factor Q$ is the inverse of
the fraction of the tidal energy that is dissipated per tidal cycle. You can assume that Q$ = 11.5,
the value seen today.
The present day value of the Lunar semimajoraxis is a$(0) = 384, 000 km.
The Solar tidal torque, T , is usually taken to be proportional to TL, a practice we will
follow: TL + T = TL(1 + β), where
β =

. (5)
The factor 1/4.7 is the present day ratio of T /T$; since a$/a$(0) was less than one in the
past, the Solar tide was relatively less important then.
This is in initial value problem, if you think of time starting at t = 0 and becoming more
and more negative, i.e., t < 0. You can use the integrator of your choice; I would suggest either
odeint or solve ivp.
Since it is an initial value problem, you will need the initial values:
L⊕ = M⊕
G(M + M⊕)a⊕, (6)
S⊕ = IΩ⊕, (7)
L$ = m$
G(M⊕ + m$)a$. (8)
The mass of the sun is M = 1.98 × 1033 g, that of Earth is M⊕ = 5.97 × 1027 g, the
semimajor axis of Earth’s orbit is a⊕ = 1.49 × 108 km, and the Earth’s moment of inertial is
I = 0.3299M⊕R2
⊕. The angular velocity of the Earth is Ω⊕ = 2π/lod, where lod is the length
of the sidereal day, 86164 seconds (about four minutes less than the length of the solar day, 24
hours, by definition).

  1. Pick a unit system. I suggest cgs, since I gave most–but not all!–of the quantities in cgs.
    Calculate L⊕, S⊕ and L$ in those units and report them. (10 pt)
  2. Give the present day values of T$ and T⊕ in the same unit system. You can check this
    against values you can find, but if you do, report where you found them. (10 pt)
  3. Calculate the three timescales associated with equations (1)-(3), in years; for example,
    from equation (1)
    τL⊕ = L⊕/T . (9)
    (10 pt)
  4. Write a function to evaluate the right hand sides of equations (1)-(3). This should look like
    either “damped oscillator” or “pendulum” in the numerical integration.ipynb notebook
    from Friday’s lecture. Note that those examples are very terse, and yours might be a bit
    longer. You might also want to define auxiliary functions like T moon(a moon). (10 pt)
  5. Use this function, together with odeint or solve ivp, to integrate back in time until the
    Moon hits Earth. How long, in years, did the Moon form, according to this tidal model?
    (20 pt)
  6. Make a figure showing a$(t). Label the axes, with units. I suggest either millions or
    billions of years for the x-axis (age), and kilometers for the y-axis. Note that even if
    you get an answer for question 5 that you find unreliable, you will still get full credit for
    making the figure. (10 pt)
  7. Make a figure showing the length of the day (in hours) versus age, with the same note as
    in the previous question. (10 pt)
  8. It is believed that the Moon formed just outside the Roche radius. Look up the Roche
    radius, and, assuming that the moon did form there, what was the length of the day at the
    time of the Moon’s formation? (10 pt)
  9. Look up estimates for the age of the Moon, and of the Earth, and report them. You should
    find that both the Earth and the Moon are older than the estimate from the tidal equations.
    This problem was first noted in the 1950’s. (10 pt)
  10. BONUS: Try to think of what might be wrong, either in the tidal equations, or in the
    estimates of the age of the Earth and the Moon. Then order them by what you think is the
    most likely problem, and give reasons for your ordering. (10 pt)

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