Specific Mechanical Energy (1)

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Equation / Last modified by KurtHeckman on 2018/05/23 19:13
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MichaelBartmess.Specific Mechanical Energy (1)
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This Specific Mechanical Energy equation computes the mechanical energy per unit mass ) as a constant characterizing a circular satellite orbit.  This equation uses the Earth Gravitational Constant, `mu`, and the input length of the orbital radius of the satellite orbit, `r`. 

INSTRUCTIONS: Choose the preferred units and enter the following:

  • (a) This is the radius of the satellite's orbit

Specific Mechanical Energy of the Satellite (ε): The calculator returns the energy per unit mass of a satellite in a circular orbit of the specified radius. The result is in units of Grays, where  1 Gy = 1 `"Joule"/"kilogram"` = 1 `"meter"^2`/`"sec"^2`

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The Math / Science

The formula for the Specific Mechanical Energy of a Satellite is:

`ε = -μ/(2*r)`

where:

Derivation

The mechanical energy of an object is a sum of its kinetic energy (K) and potential energy (U):

(1) `E = K + U`

The kinetic energy of satellite can be written using the equation:

(2) K = ½⋅m⋅v², where `m` is the satellite's mass and `v` is the satellite's velocity

And the potential energy of a satellite can be written using the equation:

(3) `U = -(G*M*m)/r`, where `G` is the universal gravitational constant, `M` is the mass of the Earth,

Therefore, substituting (2) and (3) into equation (1), we get:

(4) `E = 1/2 * m * v^2 - (G*M*m)/r`

In a circular orbit, Newton's 2nd Law dictates:

(5) `(G*M*m)/r^2 = (m*v^2)/r`

And multiplying both sides of (5) by `r` we get:

(6) `(G*M*m)/r = (m*v^2)`

Using (6) and substituting for `(m*v^2)`  (4) becomes:

(7) `E = 1/2 * m * v^2 - (G*M*m)/r  = 1/2 * (G*M*m)/r - (G*M*m)/r = -(G/2)*(M*m)/r` 

Setting `mu` to be the Earth's gravitational constant:

(8) `mu = G*M`

We get from (7):

(9) `E = -mu/(2*r) * m`

And by definition the specific mechanical energy of the satellite is the mechanical energy per unit mass:

(10) `epsilon = E/m = (-mu/(2*r) * m)/m`

And finally we get:

(11) `epsilon = -mu/(2*r)`