Introduction
This is the second of three arguments that explain the origin of moons: capture. As previously stated, with the exception of our moon, few theories exist that explain how they got there. Most that do, rely on some assembly from a disk of smaller particles. Some mention a possibility of capture — although capturing a moon can be difficult.
Here, we continue an argument producing these formations from liquids. We have shown that an impact into a liquid planet can project a liquid from the surface in the form of a stream. We now show how streams produce a capture mechanism.
The streams, or projections, are divided into two parts – the main body with enough mass to gravitationally attract (an earth or larger) and a lower mass tail (a moon). Here, the main body of the stream can influence a capture of the “tail” into an orbit if the source is spinning.
Projections (streams) from a source are divided energetically. Higher energy projections may escape, or return and impact, as an orbiting pair – where the main body captures the tail. Lower energy projections offer a mechanism where the source captures the moon, using the main body of the stream as a surrogate. Even lower energy projections that stay close to the surface, may offer an origin of Earth’s moon.
Simulation of Orbits
The projections (streams) are modeled as a three body problem with the ODE simulator in Python, odeint(f,r,t). The function, f, contain the equations of gravitational force from the source and between the planet and moon. The function processes the positions and velocities of the planet and moon contained in the variable, r. The following represents this main process:
x1 = r[0] y1 = r[1] xdot1 = r[2] ydot1 = r[3]
x2 = r[4] y2 = r[5] xdot2 = r[6] ydot2 = r[7]
d1 = sqrt{x1**2+y1**2} d2 = sqrt{x2**2+y2**2}
d = sqrt{(x1-x2)**2+(y1-y2)**2}
xddot1 = -m*g*x1/d1**3 yddot1 = -m*g*y1/d1**3
xddot2 = -1.0*m*g*x2/d2**3 - 1.0*m1*g*(x2-x1)/d**3
yddot2 = -1.0*m*g*y2/d2**3 - 1.0*m1*g*(y2-y1)/d**3
where 1 is the planet and 2 is the moon
The initial positions and velocities are assigned to r as follows:
x1 = r + n*Rs
y1 = 0.0
vx1 = sqrt{2*g*m*(1/x1 - 1/rs)}
vy1 = sqrt{g*m/rs}
x2 = r
y2 = 0.0
vx2 = sqrt{2*g*m*(1/x2 - 1/rs)}
vy2 = sqrt{g*m/rs}
where,
r is the radius of the sun Rs is the radius of the planet (Saturn, in this case) rs is the distance from the planet n is a multiple for convenience
As a brief explanation, we model the stream as a two body, projected from some larger source (Sun, Saturn…). The first body, x1, about ninety percent of the mass, is projected a distance, n*Rs, from the surface. The second body, x2, is then projected and “follows” x1, which simulates the stream. Both will follow an elliptical orbit to some apse point, where both objects begin to return.
In many cases, the tail, x2, will “whip” around, where x1 can capture x2 in an orbit. x1 will eventually crash into the source, but x2 may not and begin to orbit the source. In other words, x2 can use x1 as a surrogate to obtain an orbit around the source.
A Problem With Projections
Orbits require two thrusts – one for height and one in the direction of the orbit. Impacts from the outside do not have this capability.
As an example, the orbits on the left in Fig. 1 are produced by an object placed one earth radii above the surface. The object is then given different tangent velocities (think Newton’s cannon). The height and direction are defined. This can obtain an orbit. The outer orbit is one type we seek.
The orbits on the right in Fig. 1 are produced by an object placed on the surface. This may simulate the debris from a solid (or liquid) impact. Using the same spin velocities, the object is given a radial velocity necessary to reach one earth radii above the surface. Free from the surface, the velocity of the object assumes the sum of its components — radial and spin. No orbit is obtained. Thus, a projection from the surface of a spinning planet, on its own, is destined to fall back to it.
The problem is the geometry of the projectile. If a liquid is projected as a stream, modeled as a two-body, the main body of the stream will attract the smaller tail, if it has enough mass. This capture mechanism is consistent in numeric simulation (see above).
Higher Energy Projections
In high energy projectiles from the Sun, or a larger planet, the main body of the stream captures the moon (its tail). Both may return (impact) or escape as a pair. The mechanism shown in Fig. 2 demonstrates the gravitational attraction of a stream the mass of Jupiter, projected from the Sun. There is 50 Jupiter radii between the head (Jupiter) and the tail (moon). Plots look similar for a projected Saturn at 40 radii, a projected Earth at 5 radii or a projected Mars at 1 radii.
The plot in Fig. 3 demonstrates the influence of stream length. The top shows a less stable capture (orbit), at 10 Jupiter radii length stream, from top (planet) to bottom (moon) – and a more stable capture on the bottom, at 100 Jupiter radii length. (Note: the sun can project from impacts and explosions.)
The plot in Fig. 4 shows a captured moon from a projected “earth” from a liquid Saturn. The earth/moon pair will escape, or impact back into Saturn, as a pair.
Lower Energy Projections
As a rough definition, a lower energy projection will detach from the surface of the source, but not far enough for the main mass to capture the moon. Instead, it acts as a surrogate for the source, Saturn in this case, as shown in Fig. 5. To demonstrate the mechanism, place the moon at the surface and an “earth” at r_{\textrm{s}} (left) and 2r_{\textrm{s}} (right) above the surface. Once the “earth” (red) crashes back into the planet, it is removed from the calculations (m_{\textrm{e}} = 0) — so disregard red orbits. But the moon (blue) does obtain an orbit. This is how most moons are captured.
Same simulation equations as above – just replace Saturn for the Sun. I use Earth as the surrogate because it has enough mass to capture a moon. As a stream, “earth” gets a head start over the “moon”, so it is placed above the surface to start, with the appropriate velocity. Saturn spin is 10 Km/sec (the current spin) and a radius of 30,000 Km.
The Moon
How did our moon get in orbit? The giant impact theory is currently popular, and a good overview can be seen here^{1}. From the article, the chemical content of the moon and earth are similar — meaning the material of the moon comes from the earth and not from the impactor. This would explain the chemical content. But a projectile from the earth will be less massive than the earth. It will not capture the tail. A different mechanism for earth is needed.
If the projected stream rotates with the earth as one piece while attached – it will not act as a free particle. It retains its angular velocity and offers the possibility of an orbit — like the outer orbit shown on the left in Fig. 1. This can happen if the spin of the earth matches the orbital velocity at the top of the stream (at r_{\textrm{e}} above the surface, say). It is very possible the earth was spinning this fast when it first formed.
An impact into a liquid (earth) produces a projection in the form of a stream. If the stream stays close to the surface, it will rotate with the earth as one piece while attached – it will not act as a free particle. It retains its angular velocity and offers the possibility of an orbit.
This is simply a form of Newton’s cannon. This happens if the spin velocity of the earth can place an object near the surface in orbit. This seems probable if the spin of an early Earth was close to its orbital speed, similar to Jupiter and Saturn. A demonstration of the mechanism is shown in Fig. 7.
Conclusion
We have shown several mechanisms that capture moons into an orbit. We now show how these moons migrate into the circular orbits we see today.
Please read the argument on the migration of moons, next.
_____________________________________________________________________________ ^{1}https://astronomy.com/news/2019/05/giant-impact-hypothesis-an-evolving-legacy-of-apollo