M079 · Hyperbolic encounter / design

Gravity-Assist Mission Designer

Aim a hyperbolic encounter around an Earth or Jupiter reference. Drag the periapsis, reverse the turn and switch reference frames; connect the real trajectory to asymptotic velocity triangles, energy exchange and a safe outgoing-speed target.

Time from periapsis $\text{Pending}$

Body reference $\text{Pending}$

Asymptotic turn angle $\text{Pending}$

Hyperbolic eccentricity $\text{Pending}$

Planned periapsis altitude $\text{Pending}$

Impact parameter / body radius $\text{Pending}$

Incoming inertial speed $\text{Pending}$

Outgoing inertial speed $\text{Pending}$

Signed inertial speed change $\text{Pending}$

Asymptotic energy exchange $\text{Pending}$

Current planet-frame speed $\text{Pending}$

Periapsis speed $\text{Pending}$

Safety check $\text{Pending}$

Signed outgoing speed error $\text{Pending}$

Speed target check $\text{Pending}$

Sampled energy defect $\text{Pending}$

Sampled momentum defect $\text{Pending}$

Sampled propagation residual $\text{Pending}$

OBSERVE / MEASURE / DESIGN

How can a flyby change speed without an engine burn?

Body: Earth / Jupiter

b

0.00\,

Incoming excess speed

v_\infty

6.00\,km/s

Approach direction

\alpha_{\rm in}

90.00\,{}^\circ

Periapsis radius

r_p/R_p

1.30\,

Turn sign

s

-1.00\,

Safety radius

r_{\rm safe}/R_p

1.10\,

Outgoing inertial speed target

V_{\rm target}

36.00\,km/s

Frame: planet / inertial

f

0.00\,

Reviewed time

t/t_{\rm obs}

0.00\,\%

The orange periapsis ring edits closest approach in either frame; every drag has a keyboard slider. The dashed red circle travels with the planet and marks the chosen safety margin. The speed target compares the outgoing inertial asymptote with a tolerance of a quarter kilometre per second; it imposes no destination or direction constraint. Gray target and purple trade curve show that lower is not always better.

Planar unpowered patched-conic encounter with a spherical body, constant planet velocity and negligible spacecraft mass. No atmosphere, tides, oblateness, solar acceleration or ephemeris. The plotted coast has a finite observation boundary; asymptotic vectors are evaluated at infinity. Surface contact stops the path and blocks outgoing claims. Safety altitude is a chosen design margin, not a certified atmospheric boundary.

Start with the essentials

Focus question: Can one unpowered encounter gain or lose speed depending on which side it passes?
One-sentence intuition: Rotation preserves the planet-relative asymptotic speed. Adding a moving velocity to two differently directed vectors changes their inertial magnitudes and transfers orbital energy.

Core mathematical model

The same velocity in two frames

\mathbf V_{\rm in}=\mathbf V_p+\mathbf v_{\infty,\rm in},\qquad\mathbf V_{\rm out}=\mathbf V_p+\mathbf v_{\infty,\rm out}

Capital velocities refer to inertial coordinates; excess vectors are planet-relative asymptotes. The planet velocity is prescribed constant throughout this local encounter.

A hyperbola fixes the turn

e_h=1+\frac{r_pv_\infty^2}{\mu_p},\qquad\delta=2\arcsin\frac{1}{e_h}

For fixed incoming excess speed, a closer periapsis gives a stronger deflection. The signed turn selects the side of the planet.

Aim with an impact parameter

b=r_p\sqrt{1+\frac{2\mu_p}{r_pv_\infty^2}},\qquad v_p=\sqrt{v_\infty^2+\frac{2\mu_p}{r_p}}

The impact parameter is the incoming asymptote offset, not the closest distance. Local periapsis speed exceeds the asymptotic speed.

Unpowered excess speed is conserved

|\mathbf v_{\infty,\rm out}|=|\mathbf v_{\infty,\rm in}|,\qquad\varepsilon_p=\frac{v^2}{2}-\frac{\mu_p}{r}=\frac{v_\infty^2}{2}

Planet-frame speed rises then falls along the finite trajectory. Only its two asymptotic magnitudes are equal.

Where the exchanged energy comes from

\Delta\varepsilon_{\rm inertial}=\frac{V_{\rm out}^2-V_{\rm in}^2}{2}=\mathbf V_p\cdot(\mathbf v_{\infty,\rm out}-\mathbf v_{\infty,\rm in})

The moving planet supplies or absorbs orbital energy. Its recoil is negligible only because spacecraft mass is neglected; the turn can also reduce inertial speed.

Reach a safe scalar speed target

r_p>R_p,\quad r_p\ge r_{\rm safe}\ge R_p,\qquad |V_{\rm out}-V_{\rm target}|<0.25\,\mathrm{km/s}

This educational design checks one speed and a chosen margin. It does not solve a destination transfer. Contact at or within the body surface terminates propagation analytically.

Common difficulties

A faster exit needs a frame

Typical misconceptionGravity increases incoming and outgoing planet-relative speed.

Better mental modelThe two excess speeds are equal. Adding the moving planet velocity changes the inertial speed.

Closer is not always better

Typical misconceptionThe lowest allowable altitude always maximizes speed gain.

Better mental modelThe energy exchange is a vector dot product. Too much rotation can overshoot the direction that maximizes the outgoing speed.

Finite boundaries are not infinity

Typical misconceptionThe first plotted speed equals the excess speed.

Better mental modelGravity still affects both finite boundary states. The triangle uses asymptotes, while the history uses local speeds.

An impact has no valid outgoing flyby

Typical misconceptionA mathematical hyperbola can pass through the planet and still count as success.

Better mental modelThe first surface intersection stops the record and blocks outgoing results. A separate user-selected safety margin can reject a non-impacting path.

Run the experiment

01
Compare frames
Acquire the gain preset, then select inertial coordinates.
What to observe: The same encounter retains its planet-frame energy while the moving planet changes inertial velocity.
02
Reverse the exchange
Compare the gain and loss presets at the same radius.
What to observe: The excess speed is unchanged, but the outgoing inertial magnitude responds to turn direction.
03
Design safely
Drag periapsis and adjust the outgoing scalar speed target; inspect the trade chart.
What to observe: Only safe non-impacting encounters may pass the speed target. The trade curve may have an interior maximum.
04
Stop an impossible mission
Compare the unsafe-margin and surface-interception presets.
What to observe: One trajectory misses the chosen margin; the other stops at physical contact and has no valid outgoing result.