Manoeuvres

• Combine non-collinear burns (vector law / law of cosines for velocities)
  • Use cases
• Hohmann
  • Essentials
  • LEO → GEO Hohmann transfer (raise)
  • Higher → Lower circular orbit (retrograde burns)
  • Degenerate case (no change): r1 == r2
  • Use cases
• Oberth effect (specific energy gain near periapsis)
  • Use cases

Combine non-collinear burns (vector law / law of cosines for velocities)

Example usage of combineBurnsDeltaV.

import { combineBurnsDeltaV } from '@interstellar-tools/equations';
import type { Radians } from '@interstellar-tools/types';

// Helper: degrees → branded radians
const toRadians = (deg: number): Radians => ((deg * Math.PI) / 180) as Radians;
// Two burns (magnitudes in m/s)
const v1 = 500; // first impulse
const v2 = 300; // second impulse
// 1) Same direction (Δθ = 0°) → Δv = |v1 - v2|
const dv_same = combineBurnsDeltaV(v1, v2, toRadians(0));

console.log('Δθ = 0°   → Δv =', dv_same.toFixed(3), 'm/s');

// 2) Opposite directions (Δθ = 180°) → Δv = v1 + v2
const dv_opp = combineBurnsDeltaV(v1, v2, toRadians(180));

console.log('Δθ = 180° → Δv =', dv_opp.toFixed(3), 'm/s');

// 3) General non-collinear case (e.g., Δθ = 60°)
const dv_60 = combineBurnsDeltaV(v1, v2, toRadians(60));

console.log('Δθ = 60°  → Δv =', dv_60.toFixed(3), 'm/s');

// 4) Another example (Δθ = 25°)
const dv_25 = combineBurnsDeltaV(1200, 800, toRadians(25));

console.log('Δθ = 25°  → Δv =', dv_25.toFixed(3), 'm/s');

Use cases

• Combine tangential + normal at a node When a manoeuvres requires changing speed (tangential) and doing a small plane change (normal) in the same burn, set v1 and v2 to those component magnitudes, and ($\mathrm{\Delta }\theta$) (deltaTheta) to the angle between their directions. combineBurnsDeltaV returns the single-impulse equivalent $\mathrm{\Delta }v$ for budgeting and comparison.

• Hohmann + plane change at apogee/perigee During a Hohmann transfer you often circularize (tangential) and trim inclination (normal) at the same point. Use this to compute the combined burn magnitude instead of summing scalars. Tip: at apogee (lower speed) the plane-change component costs less; reflect that in v1, v2.

• Rendezvous cleanup: along-track + cross-track trim Proximity ops frequently need a small in-plane correction and a tiny out-of-plane correction. Treat them as two non-collinear components to get the single $\mathrm{\Delta }v$ you'll command to guidance.

• Dogleg + insertion If ascent/insertion includes a heading change (dogleg) and a circularization impulse, the thrust axes are not aligned. Use the mutual angle to compute the net $\mathrm{\Delta }v$ penalty of doing both together.

• Mid-course correction vectoring Interplanetary MCCs are often specified as components (R/T/N). If you execute two components in one firing with a known separation angle, get the resultant $\mathrm{\Delta }v$ to validate the manoeuvres plan.

• Thruster gimbal limits / split burns If attitude or gimbal constraints force two impulses separated by a known ($\mathrm{\Delta }\theta$), compute the $\mathrm{\Delta }v$ impact vs. the ideal pointing case.

• Bounds & sanity checks Quickly verify plans against the vector-law bounds: ($|v_1-v_2|\le \mathrm{\Delta }v\le v_1+v_2$), with extremes at ($\mathrm{\Delta }\theta =0$) and ($\mathrm{\Delta }\theta =\pi$). Useful in reviews and automated validation.

• What-if sweeps Fix v1 (e.g., circularization) and vary v2 and $\mathrm{\Delta }\theta$ (e.g., plane-change size and placement) to find cost-efficient compositions. This helps decide where to perform a plane change (apogee vs. perigee) or how much to defer.

Guidance:

• Use radians for deltaTheta ($0\dots \pi$).
• If your "burns" are orthogonal components you plan to fire simultaneously, $\mathrm{\Delta }\theta \approx {90}^{\circ }$.
• If you fire back-to-back in the same direction, $\mathrm{\Delta }\theta =0^{\circ }$ ($\mathrm{\Delta }v$ reduces to $|v_1-v_2|$ if one partly cancels the other).
• If they oppose, $\mathrm{\Delta }\theta ={180}^{\circ }$ (Δv sums to $v_1+v_2$).

Hohmann

Example usage of: hohmannTransfer, hohmannSemiMajorAxis, hohmannTransferTime.

Essentials

import {
  hohmannTransfer,
  hohmannSemiMajorAxis,
  hohmannTransferTime
} from '@interstellar-tools/equations';

// Earth's standard gravitational parameter (μ), m^3/s^2
const muEarth = 3.986004418e14;

LEO → GEO Hohmann transfer (raise)

const rLEO = 6378e3 + 400e3; // Earth radius + 400 km (m)
const rGEO = 42164e3; // GEO radius (m)
const raise = hohmannTransfer(rLEO, rGEO, muEarth);

console.log('LEO → GEO');
console.log('at (m)        :', raise.at.toFixed(0));
console.log('Δv1 (m/s)     :', raise.dv1.toFixed(2), raise.dir1); // prograde
console.log('Δv2 (m/s)     :', raise.dv2.toFixed(2), raise.dir2); // prograde
console.log('ΔvTotal (m/s) :', raise.dvTotal.toFixed(2));
console.log('tTransfer (h) :', (raise.tTransfer / 3600).toFixed(2));

// Optionally via convenience helpers:
const at = hohmannSemiMajorAxis(rLEO, rGEO);
const tTransfer = hohmannTransferTime(rLEO, rGEO, muEarth);

console.log('at (from helper)       :', at.toFixed(0));
console.log('tTransfer (from helper):', (tTransfer / 3600).toFixed(2));

Higher → Lower circular orbit (retrograde burns)

const rHigh = 12000e3; // m
const rLow = 7000e3; // m
const lower = hohmannTransfer(rHigh, rLow, muEarth);

console.log('\nHigh → Low');
console.log('Δv1 (m/s) :', lower.dv1.toFixed(2), lower.dir1); // retrograde
console.log('Δv2 (m/s) :', lower.dv2.toFixed(2), lower.dir2); // retrograde
console.log('ΔvTotal   :', lower.dvTotal.toFixed(2));

Degenerate case (no change): r1 == r2

const rSame = rLEO;
const noop = hohmannTransfer(rSame, rSame, muEarth);

console.log('\nNo-op (same orbit)');
console.log('Δv1, Δv2, ΔvTotal:', noop.dv1, noop.dv2, noop.dvTotal); // all ~0
console.log('dirs:', noop.dir1, noop.dir2); // both 'none'
console.log(
  'tTransfer (half period):',
  (noop.tTransfer / 60).toFixed(2),
  'min'
);

Use cases

• Raise LEO → GEO (classic design case) Compute the budget for a coplanar circular raise from ($r_1$) (e.g., LEO) to ($r_2$) (GEO):

  • Transfer semi-major axis: ($a_t=\frac{r_1+r_2}{2}$)
  • Burns: ($\mathrm{\Delta }{v}_1=\sqrt{\frac{\mu }{r_1}}(\sqrt{\frac{2r_2}{r_1+r_2}}-1),\mathrm{\Delta }{v}_2=\sqrt{\frac{\mu }{r_2}}(1-\sqrt{\frac{2r_1}{r_1+r_2}})$)
  • Total: ($\mathrm{\Delta }{v}_{\text{total}}=\mathrm{\Delta }{v}_1+\mathrm{\Delta }{v}_2$)
  • Time of flight: ($t_t=\pi \sqrt{\frac{a_t^3}{\mu }}$) (half the transfer ellipse period) Use the returned dir1/dir2 to label burns as prograde/retrograde for ops planning.

• Lowering orbit (deorbit preparation or constellation altitude change) For ($r_2<r_1$), the same formulas apply; the API returns magnitudes ($\mathrm{\Delta }{v}_1,\mathrm{\Delta }{v}_2$) with directions:

  • First burn is retrograde at ($r_1$) (speed decrease),
  • Second burn is retrograde at ($r_2$) to circularize. This is useful for controlled perigee lowering or moving satellites to a new operational shell.

• Time-constrained transfers When you must meet a window, use ($t_t=\pi \sqrt{\frac{a_t^3}{\mu }}$) to check feasibility against ground-station coverage, lighting constraints, or rendezvous timing. If ($t_t$) is too long, you may need a higher-energy (non-Hohmann) profile.

• Budgeting & trades against bi-elliptic For large radius ratios, a bi-elliptic transfer can beat Hohmann in total ($\mathrm{\Delta }v$). Hohmann is often optimal up to about ($\frac{r_2}{r_1}\approx 11.94$) (rule of thumb). Use this function to:

  • Rapidly compute the Hohmann baseline ($\mathrm{\Delta }{v}_{\text{total}}$),
  • Compare with a bi-elliptic design if ($\frac{r_2}{r_1}$) is very large.

• "No-op" sanity check If ($r_1=r_2$), then ($a_t=r_1$) and the function returns ($\mathrm{\Delta }{v}_1=\mathrm{\Delta }{v}_2=0$) with ($t_t=\pi \sqrt{\frac{r_1^3}{\mu }}$) (half the circular period). This is handy in tests and CI to validate inputs and units.

• Combining with plane change at apogee Hohmann is coplanar. If a small inclination change is needed, perform it where the speed is lowest (typically at apogee). After computing the tangential circularization ($v$)-change at ($r_2$), combine it with a normal component using your plane-change tools, e.g. ($\mathrm{\Delta }{v}_{\text{combined}}=\sqrt{v_t^2+v_n^2-2v_t{v}_n\cos \mathrm{\Delta }\theta }$). This gives a more realistic single-burn magnitude for the apogee manoeuvre.

• Constellation shell changes and station-keeping resets Use ($\mathrm{\Delta }{v}_{\text{total}}$) and ($t_t$) to estimate fuel and downtime when migrating satellites between shells (e.g., ($550\text{km}\to 600\text{km}$)), or when re-baselining after long station-keeping campaigns.

Guidance

• Inputs: ($r_1,r_2$) in meters, ($\mu$) in m(${}^3$)/s(${}^2$).
• Outputs: ($\mathrm{\Delta }{v}_1,\mathrm{\Delta }{v}_2,\mathrm{\Delta }{v}_{\text{total}}$) in m/s; ($t_t$) in seconds.
• Works for both raise ($(r_2>r_1)$) and lower ($(r_2<r_1)$).
• Use returned dir1/dir2 to map to prograde/retrograde/none for burn scripting.

Oberth effect (specific energy gain near periapsis)

Example usage of oberthEnergyGain.

import { oberthEnergyGain, circularSpeed } from '@interstellar-tools/equations';

// Earth’s GM (μ) in m^3/s^2
const muEarth = 3.986004418e14;
// Example: small prograde burn at LEO periapsis (~400 km altitude)
const rLEO = 6378e3 + 400e3; // m
const vLEO = circularSpeed(rLEO, muEarth); // m/s ≈ 7670
// Suppose guidance commands a small impulsive burn Δv = 50 m/s at periapsis
const dv = 50; // m/s
// Oberth approximation: Δε ≈ v · Δv  (specific energy gain, J/kg ≡ m^2/s^2)
const deltaEps = oberthEnergyGain(vLEO, dv);
// For intuition, translate Δε to an approximate change in semi-major axis (elliptic case):
// ε = -μ/(2a)  ⇒  Δa ≈ (a^2 / μ) · Δε (valid for small changes around circular LEO)
const aLEO = rLEO; // circular → a = r
const deltaA = ((aLEO * aLEO) / muEarth) * deltaEps; // meters

console.log(`LEO speed v ≈ ${vLEO.toFixed(1)} m/s`);
console.log(`Prograde Δv  = ${dv.toFixed(1)} m/s`);
console.log(`Specific energy gain Δε ≈ ${deltaEps.toFixed(0)} J/kg`);
console.log(`Approx. semi-major axis change Δa ≈ ${deltaA.toFixed(0)} m`);

// Tip: If the burn is not perfectly prograde by angle φ (in radians),
// scale by cos φ: Δε ≈ v · Δv · cos φ  (small-angle approximation).

Use cases

• Maximize energy gain at periapsis (departure burn) When planning an escape or high-energy transfer, compute the specific energy increment from a periapsis burn where orbital speed ($v$) is highest: ($\mathrm{\Delta }\epsilon \approx v\mathrm{\_}\text{peri}\mathrm{\Delta }v$). Use this to show why concentrating ($\mathrm{\Delta }v$) near periapsis is more effective than the same ($\mathrm{\Delta }v$) elsewhere.

• Compare burn placement: periapsis vs. apoapsis For the same impulsive ($\mathrm{\Delta }v$), evaluate the energy benefit at different true anomalies: ($\mathrm{\Delta }\epsilon \ast \text{peri}\approx v\ast \text{peri}\mathrm{\Delta }v$) vs. ($\mathrm{\Delta }\epsilon \ast \text{apo}\approx v\ast \text{apo}\mathrm{\Delta }v$) with ($v\ast \text{peri}\gg v\ast \text{apo}$). Use this to justify performing plane changes or trims at apoapsis (cheap angle change) but energy-raising burns at periapsis.

• Interplanetary departure staging (parking → hyperbolic) Budget the hyperbolic excess energy by applying the Oberth estimate at LEO/LEO-like periapsis: ($\mathrm{\Delta }\epsilon \approx v\mathrm{\_}\text{LEO}\mathrm{\Delta }v$). Helps compare "all at LEO" vs. "split between LEO and deep-space" strategies.

• Powered gravity assist (periapsis burn during flyby) During a close planetary flyby, add a small prograde burn near pericenter to leverage the high ($v$) for a large energy increase: ($\mathrm{\Delta }\epsilon \approx v\mathrm{\_}\text{p}\mathrm{\Delta }v$). Useful to illustrate the payoff of perijove/periapsis burns in Jovian assists.

• Mission design trade: deep-space manoeuvre vs. low-periapsis burn For a fixed ($\mathrm{\Delta }v$) capability, compare: ($\mathrm{\Delta }\epsilon \ast \text{DSM}\approx v\ast \text{DSM}\mathrm{\Delta }v$) vs. ($\mathrm{\Delta }\epsilon \ast \text{peri}\approx v\ast \text{peri}\mathrm{\Delta }v$). Shows why moving thrust "down the gravity well" can reduce total ($\mathrm{\Delta }v$) to reach a target energy.

• Angle-off-velocity corrections (non-prograde burn) If the burn is not perfectly aligned with velocity by angle ($\varphi$), use the small-angle form to estimate the effective energy gain: ($\mathrm{\Delta }\epsilon \approx v\mathrm{\Delta }v\cos \varphi$). Demonstrates pointing sensitivity and why guidance strives for prograde alignment during energy-critical burns.

• Back-of-the-envelope fuel/energy checks Quickly convert available ($\mathrm{\Delta }v$) into expected specific energy change at a known speed ($v$): ($\mathrm{\Delta }\epsilon \approx v\mathrm{\Delta }v$). Handy for sanity checks in ops consoles, design docs, and CI tests without invoking full orbit propagation.

Notes

• Valid for small, impulsive, prograde burns; higher-order term ($\frac{1}{2}(\mathrm{\Delta }v{)}^2$) is neglected.
• Use with care when ($\mathrm{\Delta }v$) is large or burn duration is non-negligible.