specific orbital energy
Definition: The specific orbital energy (total mechanical energy per unit reduced mass) of a bound two-body orbit is ε = -GM/(2a), a constant of the motion that depends only on the semimajor axis a, not on the eccentricity or the instantaneous position in the orbit.
Why it matters here: The vis-viva equation converts the single constant of orbital energy into the instantaneous speed at any point in an orbit, and chaining vis-viva calculations at two tangent orbits’ shared points is exactly how minimum-energy interplanetary transfer trajectories are designed.
Relationship: Combining this constant specific energy with the general orbital energy ε = v²/2 - GM/r at any point in the orbit and solving for v gives the vis-viva equation, v² = GM(2/r - 1/a), which reduces to the circular-orbit speed when r = a and to the escape-velocity formula when a → ∞.
Boundary or common confusion: Assuming that a Hohmann transfer is the fastest way to get from one orbit to another. It is the minimum-energy (minimum total Δv) transfer between two coplanar circular orbits, not the minimum-time transfer; faster transfers are possible but require more propellant (a larger total Δv), a tradeoff every real mission-design team must weigh explicitly.
vis-viva equation
Definition: Combining this constant specific energy with the general orbital energy ε = v²/2 - GM/r at any point in the orbit and solving for v gives the vis-viva equation, v² = GM(2/r - 1/a), which reduces to the circular-orbit speed when r = a and to the escape-velocity formula when a → ∞.
Why it matters here: The vis-viva equation converts the single constant of orbital energy into the instantaneous speed at any point in an orbit, and chaining vis-viva calculations at two tangent orbits’ shared points is exactly how minimum-energy interplanetary transfer trajectories are designed.
Relationship: A Hohmann transfer orbit is an elliptical orbit tangent to both an inner circular orbit (at periapsis) and an outer circular orbit (at apoapsis); reaching it from the inner circular orbit, and later leaving it for the outer circular orbit, each requires a single velocity change (burn) computed directly from the vis-viva equation at the transfer orbit’s periapsis and apoapsis.
Boundary or common confusion: Assuming that a Hohmann transfer is the fastest way to get from one orbit to another. It is the minimum-energy (minimum total Δv) transfer between two coplanar circular orbits, not the minimum-time transfer; faster transfers are possible but require more propellant (a larger total Δv), a tradeoff every real mission-design team must weigh explicitly.
periapsis/apoapsis
Definition: A Hohmann transfer orbit is an elliptical orbit tangent to both an inner circular orbit (at periapsis) and an outer circular orbit (at apoapsis); reaching it from the inner circular orbit, and later leaving it for the outer circular orbit, each requires a single velocity change (burn) computed directly from the vis-viva equation at the transfer orbit’s periapsis and apoapsis.
Why it matters here: The vis-viva equation converts the single constant of orbital energy into the instantaneous speed at any point in an orbit, and chaining vis-viva calculations at two tangent orbits’ shared points is exactly how minimum-energy interplanetary transfer trajectories are designed.
Relationship: The specific orbital energy (total mechanical energy per unit reduced mass) of a bound two-body orbit is ε = -GM/(2a), a constant of the motion that depends only on the semimajor axis a, not on the eccentricity or the instantaneous position in the orbit.
Boundary or common confusion: Assuming that a Hohmann transfer is the fastest way to get from one orbit to another. It is the minimum-energy (minimum total Δv) transfer between two coplanar circular orbits, not the minimum-time transfer; faster transfers are possible but require more propellant (a larger total Δv), a tradeoff every real mission-design team must weigh explicitly.
Hohmann transfer
Definition: A Hohmann transfer orbit is an elliptical orbit tangent to both an inner circular orbit (at periapsis) and an outer circular orbit (at apoapsis); reaching it from the inner circular orbit, and later leaving it for the outer circular orbit, each requires a single velocity change (burn) computed directly from the vis-viva equation at the transfer orbit’s periapsis and apoapsis.
Why it matters here: The vis-viva equation converts the single constant of orbital energy into the instantaneous speed at any point in an orbit, and chaining vis-viva calculations at two tangent orbits’ shared points is exactly how minimum-energy interplanetary transfer trajectories are designed.
Relationship: Assuming that a Hohmann transfer is the fastest way to get from one orbit to another.
Boundary or common confusion: Assuming that a Hohmann transfer is the fastest way to get from one orbit to another. It is the minimum-energy (minimum total Δv) transfer between two coplanar circular orbits, not the minimum-time transfer; faster transfers are possible but require more propellant (a larger total Δv), a tradeoff every real mission-design team must weigh explicitly.
delta-v
Definition: The specific orbital energy (total mechanical energy per unit reduced mass) of a bound two-body orbit is ε = -GM/(2a), a constant of the motion that depends only on the semimajor axis a, not on the eccentricity or the instantaneous position in the orbit.
Why it matters here: The vis-viva equation converts the single constant of orbital energy into the instantaneous speed at any point in an orbit, and chaining vis-viva calculations at two tangent orbits’ shared points is exactly how minimum-energy interplanetary transfer trajectories are designed.
Relationship: Combining this constant specific energy with the general orbital energy ε = v²/2 - GM/r at any point in the orbit and solving for v gives the vis-viva equation, v² = GM(2/r - 1/a), which reduces to the circular-orbit speed when r = a and to the escape-velocity formula when a → ∞.
Boundary or common confusion: Assuming that a Hohmann transfer is the fastest way to get from one orbit to another. It is the minimum-energy (minimum total Δv) transfer between two coplanar circular orbits, not the minimum-time transfer; faster transfers are possible but require more propellant (a larger total Δv), a tradeoff every real mission-design team must weigh explicitly.
orbital period (generalized)
Definition: The specific orbital energy (total mechanical energy per unit reduced mass) of a bound two-body orbit is ε = -GM/(2a), a constant of the motion that depends only on the semimajor axis a, not on the eccentricity or the instantaneous position in the orbit.
Why it matters here: The vis-viva equation converts the single constant of orbital energy into the instantaneous speed at any point in an orbit, and chaining vis-viva calculations at two tangent orbits’ shared points is exactly how minimum-energy interplanetary transfer trajectories are designed.
Relationship: Combining this constant specific energy with the general orbital energy ε = v²/2 - GM/r at any point in the orbit and solving for v gives the vis-viva equation, v² = GM(2/r - 1/a), which reduces to the circular-orbit speed when r = a and to the escape-velocity formula when a → ∞.
Boundary or common confusion: Assuming that a Hohmann transfer is the fastest way to get from one orbit to another. It is the minimum-energy (minimum total Δv) transfer between two coplanar circular orbits, not the minimum-time transfer; faster transfers are possible but require more propellant (a larger total Δv), a tradeoff every real mission-design team must weigh explicitly.