Prove Tidal Distortion Volume Conservation Mathematically

[cygnus2]

I know that when a spherical shell( of negligible mass) is attracted towards a massive body it distorts into an ellipse; keeping its volume const.I have also heard it said that this volume preserving property belongs to that of the inverse square law of gravity.But how do i prove it mathematically?

 

[pervect]

I know that when a spherical shell( of negligible mass) is attracted towards a massive body it distorts into an ellipse; keeping its volume const.I have also heard it said that this volume preserving property belongs to that of the inverse square law of gravity.But how do i prove it mathematically?

If you wait long enough, the ellipse will eventually become planar, so the volume isn't actually a constant - what is true is that the first and second derivatives of the volume are initially zero. (I'm getting that the third derivative is also zero)

If you assume that the shape is in fact elliptical, it isn't too hard to prove. It would take some more complicated calculations to confirm that the shape is that of an ellipse.

The volume of an ellipse (actual the surface generated by resolving an ellipse) is

4/3*pi*a*b*c

where a,b,c are the axes of the ellipsoid

The distortion force on the ellipse is just the tidal force. The tidal force has a stretching component in the radial direction, but a compressive component in the transverse directions.

The stretching force per unit distance 2M/r^3 (this is the derivative of G*M/r^2 with respect to r). The compressive forces are both m/r^3. Here M is the mass of a point mass, and r is the distance one is away from the point mass. It takes some vector diagrams to calculate the compressive tidal force, so I'll skip that step unless there is some interest in the details.

If we _assume_ that the figure takes the shape of an ellipse, we can calculate the volume simply by tracking two major axes of the ellipsoid, the one that is elongated, and the two that are shortened, as the area of an ellipsoid is just 4/3*pi*a*b*c, where a,b,c are the major axes

http://www.murderousmaths.co.uk/books/BKMM7xea.htm

The "long" axis is c(t) = c+.5*a*t^2 = c+.5*2*m/r^3*t^2 = c+t^2/r^3
The two shorter axes are a-.5*m/r^3*t^2 = a-.5*t^2/r^3

Initially we know that a=c, so we simplify the expression as

volume = (a+t^2/r^3)*(a-.5*t^2/r^3)*(a-.5*t^2/r^3)

Finding the derivatives, we get that the first derivative is proportional to t^3, so it's zero when t=0

The second derivative is proportioanl to t^2, so it's also zero.

Even the third derivative is zero, as it's proportional to 't'.'

The fourth derivative is non-zero.

 

[R0man]

To prove tidal distortion volume conservation mathematically, we can use the concept of gravitational potential energy and the principle of conservation of energy.

First, let us consider a spherical shell of negligible mass, with a radius r and a thickness dr, being attracted towards a massive body with a gravitational force F. This force can be written as:

F = GmM/r^2

Where G is the gravitational constant, m is the mass of the spherical shell, and M is the mass of the massive body.

As the spherical shell moves towards the massive body, it experiences a change in potential energy, given by:

dU = -Fdr = -GmMdr/r^2

Now, we know that the potential energy is directly related to the shape of the object, given by the equation:

U = kV

Where k is a constant and V is the volume of the object.

Since the spherical shell is being distorted into an ellipse, its volume will also change. Let us denote the new volume of the distorted shell as V', and the change in volume as dV = V' - V.

Using the relation between potential energy and volume, we can write:

dU = k(dV' - dV)

Substituting the expression for dU and rearranging, we get:

GmMdr/r^2 = k(dV' - dV)

Now, we can use the principle of conservation of energy, which states that the total energy of a system remains constant. This means that the change in potential energy must be equal to the change in kinetic energy, given by:

dU = dK

Where dK is the change in kinetic energy.

Since the spherical shell has negligible mass, we can assume that its kinetic energy remains constant during the distortion. Therefore, dK = 0 and the above equation becomes:

GmMdr/r^2 = k(dV' - dV)

Rearranging once again, we get:

dV' - dV = GmMdr/kr^2

Now, we know that the inverse square law of gravity states that the force between two objects is inversely proportional to the square of the distance between them. This can also be written as:

F = k'/r^2

Where k' is a constant.

Substituting this into our equation, we get:

dV' - dV = GmMdr/F

Since the force remains constant