{ "cells": [ { "cell_type": "code", "execution_count": 2, "id": "b8924e6d-6424-4674-a1e0-3270e3b06b4f", "metadata": { "tags": [] }, "outputs": [], "source": [ "import numpy as np\n", "import astropy.units as u\n", "import astropy.constants as c" ] }, { "cell_type": "markdown", "id": "ca3bdb1a-d44e-4fd8-b251-83b71e055805", "metadata": {}, "source": [ "# Maximum and Minimum orbital separations" ] }, { "cell_type": "markdown", "id": "108ad70e-03a9-42b8-ae99-edf7035c922c", "metadata": {}, "source": [ "In this lecture, we are going to discuss\n", "\n", "* The minimum distance between 2 bodies before disruption occurs.\n", "* The maximum separation between 2 bodies before a 3rd body is capable of disturbing the orbit." ] }, { "cell_type": "markdown", "id": "4f33f80e-e456-4458-9eaa-1c59201dae7b", "metadata": {}, "source": [ "## Roche Limit\n", "Given that tidal forces cause tidal bulges to appear in a body, then an obvious question that arises is:\"are tidal forces ever strong enough to tear a body apart?\". This is the key question that we will be covering in this next topic.\n", "\n", "The condition for a body to be disrupted due to tidal forces is that the tidal acceleration must larger than the accelration due to self-gravity. Formally, this is given by:\n", "$$\n", " a_{\\rm tidal} > a_{\\rm self-gravity}\n", "$$\n", "The **Roche limit** is the smallest orbit which is stable against this effect.\n", "\n", "### Method 1\n", "As a starting point, let's consider a small body of mass $m$ and radius $R_{\\rm m}$, which is separated from some larger, stable body of mass $M$ by a distance $d$. We want to figure out when the tidal force experienced by the small body is large enough that its self-gravity cannot resist the pull of the larger body. We can substitute in for those accelration terms using the expressions for $F_{\\rm tidal}$ and $F_{\\rm self-gravity}$. This gives\n", "$$\n", " \\frac{2GMR_{\\rm m}}{d^3} > \\frac{G m}{R_{\\rm m}^2}\n", "$$\n", "For this, we find that\n", "$$\n", " d<\\left(\\frac{2M}{m}\\right) ^{1/3} R_{\\rm m}\n", "$$\n", "So if the separation is less than this value, then the orbiting body will be torn asunder. We can rewrite this in terms of densities also (which is more convient in particular examples). **From here on out, we're going to assume a Planet-Moon system, and we'll use P for Planet and m for moon.** The densities of these bodies is given by\n", "$$\n", " M_{\\rm P} = \\rho _{\\rm p} \\frac{4}{3} \\pi R_{\\rm P}^3, \\: \\; \\; m_{\\rm m} = \\rho _{\\rm m} \\frac{4}{3} \\pi R_{\\rm m}^3\n", "$$\n", "and so\n", "$$\n", " \\frac{2M}{m} = 2 \\frac{\\rho _{\\rm P} R_{\\rm P}^3}{\\rho _{\\rm m} R_{\\rm m}^3}\n", "$$\n", "giving us\n", "$$\n", " d< 2^{1/3} \\left( \\frac{\\rho _{\\rm P}}{\\rho _{\\rm m}}\\right) ^{1/3} R_{\\rm p}.\n", "$$\n", "\n", "### Method 2\n", "Ok, the above was a very simplified way of looking at this. Let's try make it more complicated and include centripetal acceleration due to orbital motion. Consider a mass element on the side of the orbiting moon facing the planet (as below).\n", "\n", "![Tidal_Friction](Figures/Centripetal_motion.png)\n", "\n", "In this case, we need to ask the question \"when is the gravitational acceleration felt by the mass element larger than the centripetal acceleration?\"\n", "$$\n", " \\frac{G M}{(d-R_{\\rm m})^2} - \\frac{Gm}{R_{\\rm m}^2} > \\omega^2(d-R_{\\rm m})\n", "$$\n", "where $\\omega$ is the angular velocity of the moon, and is given by:\n", "$$\n", " m \\omega^2 d = \\frac{GMm}{d^2}\n", "$$\n", "$$\n", " \\omega^2 = \\frac{GM}{d^3}\n", "$$\n", "Using this expression the Roche Limit condition becomes\n", "$$\n", " \\frac{G M}{d^2} \\left( 1-\\frac{R_{\\rm m}}{d} \\right)^{-2} - \\frac{Gm}{R_{\\rm m}^2} > \\frac{GM}{d^3}(d-R_{\\rm m})\n", "$$\n", "which simplifies to (ssuming $R_{\\rm m}< \\frac{G M_1}{d^2}\n", "$$\n", "You can think of this as a tug of war. On the left, we have the acceleration felt by orbiting body ($m$) and the primary mass ($M_1$) feel due to the perburing body ($M_2$), and on the right we have the gravitational accleration the orbiting body feels towards the primary. Doing some rearranging, and assuming $d< \\frac{M_1}{d^2}\n", "$$\n", "$$\n", " d > \\left(\\frac{M_1}{2M_2}\\right)^{1/3}D\n", "$$\n", "\n", "### Example: Sun, Moon, Earth system\n", "\n", "Ok, let's set this example up as: $M_1=M_{\\oplus}=5.97\\times10^{24}$ kg, $M_2=M_{\\odot}=2\\times10^{30}$ kg, $D=1$ AU. We get\n", "\n", "$$\n", " d > \\left(\\frac{M_\\oplus}{2M_\\odot}\\right)^{1/3} D\n", "$$\n", "$$\n", " d > 1.71\\times10^9 \\; {\\rm m}\n", "$$\n", "The actual distance between the Earth and the Moon is $3.84\\times10^8$m, which is 0.22$d$. So the moon is stable against the perturbing effect of the Sun!" ] }, { "cell_type": "code", "execution_count": 7, "id": "938c71d0-b8a1-49c4-8ef5-e071aee4ad5d", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Instability limit is d>1.713132e+09 m\n" ] } ], "source": [ "M1 = 1*c.M_earth\n", "M2 = 1*u.solMass\n", "D = 1*u.au\n", "d_min = (M1/(2*M2))**(1/3)*(D)\n", "print(\"Instability limit is d>{:e}\".format(d_min.to(u.m)))" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 5 }