M074 · Orbit transfers and launch windows

Hohmann Transfer Planner

Can two burns change an orbit and meet a moving target? They can, provided you leave at the right moment. Two ideal burns carry the spacecraft out to the target's orbit along half of a transfer ellipse. The target must also reach that meeting point when the spacecraft arrives, or the two miss. The right radius, the right position along the orbit and the right velocity are separate arrival conditions.

Design a transfer between two circular orbits in the same plane. Trim both burns, delay the departure and drag the arrival circle, then compare the radius, specific energy, arrival position and velocity, and the fuel and time of a bi-elliptic alternative.

Signed first impulse $\text{Pending}$

Signed second impulse $\text{Pending}$

Elapsed $\text{Pending}$

Current radius $\text{Pending}$

Ideal total impulse $\text{Pending}$

Scheduled total impulse $\text{Pending}$

Ideal transfer time $\text{Pending}$

Arrival check $\text{Pending}$

Arrival position miss $\text{Pending}$

Arrival relative speed $\text{Pending}$

Required launch phase $\text{Pending}$

Actual launch phase $\text{Pending}$

Bi-elliptic impulse $\text{Pending}$

Bi-elliptic duration $\text{Pending}$

Sampled coast energy defect $\text{Pending}$

Sampled time residual $\text{Pending}$

Plan the burns and compare the results

Can two burns change an orbit and meet a moving target?

Departure radius

r_1

1.10\,R_\oplus

Arrival radius

r_2

6.61\,R_\oplus

Alternative apoapsis factor

r_b/\max(r_1,r_2)

3.00\,

First impulse scale

\Delta v_1/\Delta v_{1,H}

100.00\,\%

Second impulse scale

\Delta v_2/\Delta v_{2,H}

100.00\,\%

Departure delay

t_{\mathrm{wait}}

0.00\,min

Initial phase error

\delta\phi

0.00\,{}^\circ

Acquired time

t/t_{\mathrm{obs}}

0.00\,\%

Drag the orange ring below the central body to set the arrival radius; every drag has a matching keyboard slider. The initial phase assumes the craft leaves right away, so waiting changes its alignment with the target. The arrival check compares both position and velocity after the second burn. The purple trade-off curve compares ideal three-burn bi-elliptic transfers with the teal Hohmann point; it leaves out trim burns and target phasing.

The model uses an ideal spherical Earth, kilometre–second units, instantaneous burns and two-body gravity. It leaves out the atmosphere, Earth's oblateness, finite thrust and third bodies. Both circular orbits lie above the reference surface, and the trajectory ends if it touches the surface. The largest sampled coast errors and the anomaly residuals are numerical checks, not mission certification.

Physics tutorial

An orbit transfer is also a timing problem

BackgroundA short prograde burn raises the opposite side of the orbit. The spacecraft then coasts under gravity, and a second burn brings it to circular speed at the destination. A target on that destination circle keeps moving during the whole coast.

Why it mattersThe vis-viva relation and the half-ellipse travel time come from two-body energy and Kepler motion. The launch-phase calculation was inspired by NASA’s educational Hohmann construction. The bi-elliptic costs here are derived by joining two tangent half-ellipses and adding up three signed burns.

Start with the essentials

Focus question: Can a perfect change of orbital radius still miss the target?
One-sentence intuition: Yes: radius, angular position and velocity must each be matched on arrival, and waiting shifts the phase even when both burns are perfect.

Core mathematical model

Circular and elliptic speeds

v_c(r)=\sqrt{\frac{\mu}{r}},\qquad v^2=\mu\left(\frac{2}{r}-\frac{1}{a}\right)

The specific orbital energy sets the speed at each radius. Gravity is a fixed central field, and there is no thrust while coasting.

Hohmann coast time

a_H=\frac{r_1+r_2}{2},\qquad t_H=\pi\sqrt{\frac{a_H^3}{\mu}}

Both burns are scheduled at the ideal apsides. Trimming the first burn changes the actual conic, but the second burn stays at its scheduled time.

Signed burn sizes

\Delta v_{1,H}=\sqrt{\mu\left(\frac{2}{r_1}-\frac{1}{a_H}\right)}-v_c(r_1),\quad\Delta v_{2,H}=v_c(r_2)-\sqrt{\mu\left(\frac{2}{r_2}-\frac{1}{a_H}\right)}

Positive means prograde and negative means retrograde. The total cost shown adds up the sizes of the burns, not their signed values.

Phase needed at launch

\phi_{0,H}=\pi-n_2t_H\pmod{2\pi},\qquad \phi_{\mathrm{burn}}=\phi_0+(n_2-n_1)t_{\mathrm{wait}},\quad n_i=\sqrt{\frac{\mu}{r_i^3}}

The initial phase is set for departing right away. If departure is delayed, both objects keep moving on their circles before the first burn.

Energy and angular momentum change at each burn

\Delta\varepsilon=\mathbf v^-\!\cdot\!\Delta\mathbf v+\frac{\lVert\Delta\mathbf v\rVert^2}{2},\qquad\Delta h=\mathbf r\times\Delta\mathbf v

Energy and angular momentum jump at each burn and are conserved separately on each coast. For planar motion, the cross product is the single angular-momentum component.

The three-burn bi-elliptic alternative

\Delta v_B=|\Delta v_{1,b}|+|\Delta v_{b,1}+\Delta v_{b,2}|+|\Delta v_{2,b}|,\quad t_B=\pi\sqrt{\frac{a_{1b}^3}{\mu}}+\pi\sqrt{\frac{a_{b2}^3}{\mu}}

The intermediate radius lies outside both circular orbits. For large radius ratios, raising it can lower the cost, but the coast takes much longer.

Common difficulties

Reaching the right radius is not a rendezvous

Typical misconceptionOnce the spacecraft reaches the destination circle, docking is guaranteed.

Better mental modelCompare where the target actually is and the relative velocity after the burn. The launch phase matters.

A burn has a direction

Typical misconceptionLowering an orbit takes the same positive burns as raising it.

Better mental modelWhen the transfer lowers the orbit, both ideal Hohmann burns change sign.

Hohmann is cheapest only under its assumptions

Typical misconceptionHohmann is the cheapest of every imaginable transfer.

Better mental modelThe comparison covers only coplanar transfers between circular orbits under two-body gravity. At large radius ratios, a three-burn bi-elliptic transfer can cost less.

Orbits cannot pass through the Earth

Typical misconceptionA conic can simply pass through Earth.

Better mental modelThe trajectory ends where the conic first touches the reference surface, and any later burns are not carried out.

Run the experiment

01
Match the target
Acquire the nominal transfer and check the arrival distance and relative speed.
What to observe: After the second ideal burn, both are close to zero.
02
Miss even with perfect burns
Use the delayed-departure preset, then remove the delay.
What to observe: The orbital radius still matches; what changes is whether the angular positions line up.
03
Trim the burn
Use the underburn preset and look at the radius and the specific energy.
What to observe: The second burn still fires at its planned time, but now on a different actual conic.
04
Trade fuel for time
Use the large-radius-ratio preset and move the intermediate apoapsis.
What to observe: The bi-elliptic point can sit below the Hohmann cost and far to its right on the time axis.