M074 · Orbital transfer / launch window

Hohmann Transfer Planner

Design a transfer between circular coplanar orbits. Trim both impulses, delay departure and drag the arrival circle; compare radius, specific energy, arrival position and velocity, and bi-elliptic fuel–time alternatives.

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 / ACQUIRE / COMPARE

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\,\%

The orange ring below the body edits arrival radius; every drag has a keyboard slider. Initial phase assumes immediate departure, so waiting changes alignment. Arrival checks compare position and velocity after the second burn. The purple trade curve compares ideal three-burn bi-elliptic plans with the teal Hohmann point; it excludes burn trims and target phasing.

An ideal spherical Earth reference uses kilometre–second units, instantaneous impulses and two-body gravity. No atmosphere, oblateness, finite thrust or third bodies. Both circles lie above the reference surface. Surface contact ends the trajectory. Sampled coast-error maxima and anomaly residuals are numerical diagnostics, not mission certification.

Physics tutorial

An orbit transfer is also a timing problem

BackgroundA short prograde impulse raises the opposite apsis. The spacecraft then coasts under gravity, and a second impulse makes its speed circular at the destination. A target on that destination circle is moving throughout the coast.

Why it mattersThe vis-viva relation and half-ellipse timing follow two-body energy and Kepler motion. NASA’s educational Hohmann construction motivates the launch-phase calculation. Bi-elliptic costs here are derived by joining two tangent half-ellipses and counting three signed impulses.

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 are separate arrival conditions. Waiting changes phase even when both impulses remain ideal.

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)

Specific orbital energy determines speed at each radius. This is a prescribed central field, with no thrust during coasts.

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 does not move the second scheduled time.

Signed local impulses

\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 is prograde, negative is retrograde. The displayed total cost is the sum of magnitudes, not a signed sum.

Launch alignment

\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 configured for departure now. A departure delay lets both circular objects move before the first burn.

Impulse energy and momentum ledger

\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 burns; conservation applies separately to each coast. The cross product is the planar angular-momentum component.

Three-burn 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 encloses both circular orbits. Increasing it can reduce cost for large radius ratios, at the price of much longer coast time.

Common difficulties

A radius is not a rendezvous

Typical misconceptionReaching the destination circle guarantees docking.

Better mental modelCompare the target’s actual position and the post-burn relative velocity. Launch phasing matters.

An impulse has a direction

Typical misconceptionOrbit lowering needs the same positive impulses as raising.

Better mental modelBoth ideal Hohmann impulses reverse sign when the transfer lowers the orbit.

A minimum has assumptions

Typical misconceptionHohmann is cheapest among every imaginable transfer.

Better mental modelThe comparison is restricted to coplanar circular two-body transfers. A three-impulse bi-elliptic transfer may cost less at large radius ratios.

No imaginary underground coast

Typical misconceptionA conic can simply pass through Earth.

Better mental modelThe reference surface ends the acquired trajectory at its first conic contact; later impulses are not executed.

Run the experiment

01
Match the target
Acquire the nominal transfer and inspect arrival distance and relative speed.
What to observe: Both approach zero after the second ideal impulse.
02
Miss with ideal burns
Use the delayed-departure preset, then remove the delay.
What to observe: Orbital radius still matches, but angular alignment changes.
03
Trim the impulse
Use the underburn preset and inspect radius plus specific energy.
What to observe: The second burn occurs at its planned time 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 while lying far to its right in time.