This Specific Mechanical Energy equation computes the mechanical energy per unit mass (ε) 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 units and enter the following:
Specific Mechanical Energy of the Satellite (ε): The calculator returns the energy per unit mass of a satellite in Grays (Gy), where 1 Gy = 1 `"Joule"/"kilogram"` = 1 `"meter"^2`/`"sec"^2`
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The formula for the Specific Mechanical Energy of a Satellite is:
`ε = -μ/(2*r)`
where:
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)`