Re: [Phys-L] Roche limit

[John Denker <jsd@av8n.com>]

On 1/27/20 11:16 AM, Anthony Lapinski wrote:

> I'm discussing the solar system, planetary rings, Comet Shoemaker-Levy, and
> the Roche limit in my astronomy class. When a natural satellite moves less
> than 2.4 X a planet's radius, tidal forces from the planet destroy it. Does
> anyone know of a good explanation/reference that derives this number? I
> imagine it is a bit complicated for a high school class.

Well, yes, the Roche limit is slightly complicated in
the sense that the chain of reasoning requires multiple
steps.  However, each step is easy and well motivated.

The fundamental physics can be stated in a few words:
  If the tidal acceleration exceeds 1 g,
  loose material will be stripped off the satellite.

Here g denotes the satellite's own surface gravity
(not earth's gravity), i.e. the gravity coming from
its own mass.  In contrast, the tidal effects come
from the primary's mass.

So all we need to do is quantify that idea.  By now
y'all know me well enough to know what I'm going to
say next:
		SCALING LAWS!

Scaling laws are very simple yet very powerful.  They
are used in all branches of physics.  They have been
used at all times from Day One of modern science (Galileo)
up to the latest research.  They deserve more classroom
coverage than they typically get nowadays.  They are more
useful and more age-appropriate than a lot of the stuff
that is covered.

The Roche derivation goes like this

The basic gravitational field of the primary goes like
1/S^2 where S is the separation between the two bodies,
i.e. orbital radius.  This is just Newton's law of
universal gravitation.

The tide-producing field is the *difference* you get by
subtracting the basic field's value at the satellite
surface from its value at the satellite center.  This
is easy to visualize in the /accelerated/ frame that
follows the satellite center.  Using Einstein's principle
of equivalence we can zero out the acceleration at the
center, and then the surface tidal acceleration is just
the aforementioned difference.  Conceptually, this is
not complicated at all.

    (All the same words apply to the tide-producing
     physics on earth:  earth=satellite, sun=primary.)

Scaling-related mathematical fact:  The difference
between 1/S^2 and 1/(1+S)^2 scales like 1/S^3.  To
an even better approximation, 2/S^3.  If you don't
believe me, you can verify this using a calculator
or a spreadsheet;  a graph is here:
  https://www.av8n.com/physics/img48/delta-square-law.png

Scaling law:  If we replace (1+S) with (r+S) in the
denominator there, we find that the tidal acceleration
scales like r/S^3 or if you want to get fancy, 2r/S^3,
where r is the radius of the satellite.  The field is
also proportional to M, the mass of the primary.

The surface gravity of the satellite scales like m/r^2,
again in accordance with Newton's law of universal
gravitation.  Here lower-case m is the mass of the
satellite.

Plugging in, the Roche limit occurs when:
	m/M ∝ (r/S)^3

or if you want to get fancy:
	m/M = 2(r/S)^3

That's it.

==========================================

For those who are wondering how I knew to draw that graph
of 1/S^2 minus 1/(1+S)^2, or are wondering how to do the
calculation without that graph, here's the story.

This adds a few more steps to the calculation, but these
steps are easy.  Motivation:  You will see these steps
again and again in other contexts.

By way of background, calculate the square of 1.00001.
Observe that it is very close to 1.00002.  You can do
this using a calculator, or you can do it algebraically.
The mnemonic is FOIL.

More generally, (1 + ε) to the Nth power is very close to
(1 + Nε) whenever ε and Nε are both tiny compared to 1.

Physicists use this relationship all day every day.  We
do it without even noticing.  It's like breathing.  It's
called expanding to first order.  Any students who want
a deeper understanding of where it comes from can sign up
for calculus class ... but for now let's accept it as an
experimentally verified fact.

It works for negative N just as well as positive N.

Getting from 1/S^2 - 1/(1+S)^2 to 2/S^3 is now just turning
the crank.  Put the terms over a common denominator, and
apply the expand-to-first-order trick a couple of times.
It's a bunch of steps, but each step is easy.  I'll let
you fill in the details.

More generally, if you have something that scales like
S^N and you wiggle S a little bit, the effect will scale
like N S^(N-1).  That's a classic calculus result.  We
just derived it for the special case of N=-2, but it's
true more generally.

Bottom line:  The Roche limit calculation is an example
of a pattern of reasoning that physicists use all day
every day.