specific orbital energy について

), divided by the reduced mass. According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time:
:

\epsilon = \epsilon_k+\epsilon_p \!

\epsilon = } =  -}\left(1-e^2\right) = -\frac

where
*

v\,\!

is the relative orbital speed;
*

r\,\!

is the orbital distance between the bodies;
*

\mu = (m_1 + m_2)\,\!

is the sum of the standard gravitational parameters of the bodies;
*

h\,\!

is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass;
*

e\,\!

is the orbital eccentricity;
*

a\,\!

is the semi-major axis.
It is expressed in J/kg = m²·s⁻² or
MJ/kg = km²·s⁻². For an elliptical orbit the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity (parabolic orbit). For a hyperbolic orbit, it is equal to the excess energy compared to that of a parabolic orbit. In this case the specific orbital energy is also referred to as characteristic energy.
==Equation forms for different orbits==
For an elliptical orbit, the specific orbital energy equation, when combined with conservation of specific angular momentum at one of the orbit's apsides, simplifies to:
:

\epsilon = -(m_1 + m_2)\,\!

is the standard gravitational parameter;
*

\,\!

is semi-major axis of the orbit.
Proof:
::For an elliptical orbit with specific angular momentum ''h'' given by
::

= \mu p = \mu a (1-e^2)\,\!

::we use the general form of the specific orbital energy equation,
::

\epsilon=}

::with the relation that the relative velocity at periapsis is
::

=  =  =  = \,\!

::Thus our specific orbital energy equation becomes
::

\epsilon =  -  =  =  -  = \,\!

::and finally with the last simplification we obtain:
::

\epsilon = -}\,\!.

or the same as for an ellipse, depending on the convention for the sign of ''a''.
In this case the specific orbital energy is also referred to as characteristic energy (or

C_3\,\!

) and is equal to the excess specific energy compared to that for a parabolic orbit.
It is related to the hyperbolic excess velocity

v_ \,\!

(the orbital velocity at infinity) by
:

2\epsilon=C_3=v_^2\,\!.

It is relevant for interplanetary missions.
Thus, if orbital position vector (

\mathbf\,\!

) and orbital velocity vector (

\mathbf\,\!

) are known at one position, and

\mu\,\!

is known, then the energy can be computed and from that, for any other position, the orbital speed.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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