Oberth Effect

The Oberth Effect is the fact that spacecraft thrust while orbiting lower in gravity wells is more 'effective' than burns orbiting farther away.

It might seem that this is giving the spacecraft free energy, but this is of course not the case. The extra kinetic energy the spacecraft acquires is offset by the lower energy of the expelled rocket propellant being ejected lower in the gravity well's energy potential. However, since we don't really care what happens to the propellant after we expel it, it sure seems like free energy.

Real space missions make heavy use of the Oberth effect to access the outer solar system (and beyond) using a very limited amount of \(\Delta v\).

Calculator

This calculator figures the additional specific kinetic energy from the Oberth effect from an impulsive burn starting at an initial velocity.

Important notes and caveats:

We use specific energy below (i.e. energy per kilogram)
Positive velocities are prograde. Negative numbers are supported and indicate retrograde.
Real rockets act over a certain finite amount of time, not instantaneously. You must ensure that the burn time is short enough that this approximation is valid!
This calculator is not relativistically correct.
For the second part, the central object (e.g. the sun) is assumed to be much more massive than the orbiting body (e.g. spacecraft).

Velocity Before Burn:
Specific Kinetic Energy (Before):		(kinetic energy per kilogram)

Burn Velocity (\(\Delta v\)):		(impulsive burn)
Specific Kinetic Energy (Burn):		(kinetic energy per kilogram)

Velocity After Burn:
Specific Kinetic Energy (After):		(kinetic energy per kilogram)
Specific Kinetic Energy ('Extra' from Oberth):		(kinetic energy per kilogram)

For an object in an orbit with the following orbital parameters:

Central Mass \(M\):		(e.g. the mass of the Sun)
Separation at burn \(r\):		(e.g. 1 AU)
Semi-major axis \(a\):		(e.g. 1 AU)
Eccentricity (initial) \(e_0\):

Then, after burning (\(\Delta v\)) at periapsis:

Eccentricity (after burn) \(e_f\):
Semi-major axis \(a\):

Theory

In the following, we assume purely Newtonian physics and a central primary much more massive (\(M\)) than the spacecraft (\(m\)).

Oberth Effect

Specific kinetic energy (i.e., kinetic energy per kilogram) is, by definition:

\[ \epsilon_k = \frac{1}{2} v^2 \]

When we do a burn, we apply a change in velocity \(\Delta v\) to the initial velocity \(v_0\):

\begin{align*} \epsilon_k &= \frac{1}{2} \left( v_0 + \Delta v \right)^2\\ &= \frac{1}{2} v_0^2 + v_0 \Delta v + \frac{1}{2} (\Delta v)^2\\ &= \epsilon_{k,0} + v_0 \Delta v + \epsilon_{k,\Delta v} \end{align*}

The first term \(\epsilon_{k,0}\) is the specific kinetic energy we have before. The last term \(\epsilon_{k,\Delta v}\) is the specific energy we expended in the burn; if we were outside a gravity well, and therefore there were no relevant 'before' velocity (\(v_0=0\)), then this would be our total specific kinetic energy.

However, if we are moving inside a gravity well, the middle term appears. This is the Oberth effect. It's extra specific kinetic energy we got by virtue of having an initial velocity. Because the velocity of an orbit will be highest at periapsis, it follows that burns are most efficient low in a gravity well.

Eccentricity from Orbit Equation (and Vis-Viva)

The orbit equation is:

\[ r = \frac{\left\|L\right\|^2}{m^2 \mu}\left(\frac{1}{1 + e \cos(\nu)}\right) \]

Where the standard gravitational parameter is taken as \(\mu=GM\) (\(G\) is the gravitational constant and \(M\) is the mass of the central object, taken to be much larger than that of the smaller object), \(\nu\) is the true anomaly (a measure of how far we are around in the orbit), and \(L\) is the angular momentum, defined from the position vector \(\vec{r}\) and velocity vector \(\vec{v}\) as:

\[ L = m \left\| \vec{r} \times \vec{v} \right\| \]

At periapsis, \(\vec{r}\) is orthogonal to \(\vec{v}\) and \(\nu = 0\). We get:

\begin{align*} r &= \frac{ m^2 \left\|\vec{r}\right\|^2 \left\|\vec{v}\right\|^2 }{m^2 \mu} \left(\frac{1}{1 + e \cos(0)}\right) \\ &= \frac{ r^2 v^2 }{\mu} \left(\frac{1}{1 + e}\right) \\ 1 + e &= \frac{r v^2}{\mu}\\ e &= \frac{r v^2}{\mu} - 1\hspace{1cm}\text{(periapsis)} \end{align*}

Similarly, if we work through it at apoapsis, we get:

\[ e = 1 - \frac{r v^2}{\mu}\hspace{1cm}\text{(apoapsis)} \]

We can also get this in terms of the semi-major axis, by applying the vis-viva equation:

\[ v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right) \]

Combining this with the previous result will show after some algebra that:

\begin{align*} e &= \frac{2}{1+\mu/(a v^2)} - 1\hspace{1cm}\text{(periapsis)}\\ &= 1 - \frac{2}{1+\mu/(a v^2)}\hspace{1cm}\text{(apoapsis)}\\ \end{align*}

Eccentricity from Definition

The eccentricity is given by:

\[ e = \sqrt{ 1 + \frac{2 \epsilon h^2}{\mu^2} } \]

Where \(\epsilon\) is the specific orbital energy,

\begin{align*} \epsilon &= \epsilon_k + \epsilon_p\\ &= \frac{1}{2} \left\|\vec{v}\right\|^2 - \frac{\mu}{\left\|\vec{r}\right\|} \end{align*}

\(\vec{h}\) is the specific relative angular momentum (note as above that at periapsis this has magnitude just the product of the lengths of the velocity and radial vectors),

\[ \left\|\vec{h}\right\| = \frac{L}{m} = \left\|\vec{r} \times \vec{v}\right\| \]

Simplifying, we get:

\[ e = \frac{r v^2}{\mu} - 1 \]

which is the result obtained above.

Credits

This calculator was developed from a suggestion. Thanks to folks on the ToughSF Discord for helpful discussion on eccentricity.