Documentation / @interstellar-tools/equations / oberthEnergyGain

Function: oberthEnergyGain()

function oberthEnergyGain(v: number, dv: number): number;

Defined in: manoeuvres/oberth-energy-gain.ts:65

Oberth effect (specific energy gain near periapsis).

Approximation

$$\mathrm{\Delta }\epsilon \approx v\mathrm{\Delta }v$$

where:

• ($v$) is the instantaneous speed at the burn point (typically periapsis),
• ($\mathrm{\Delta }v$) is the impulsive prograde burn magnitude,
• ($\mathrm{\Delta }\epsilon$) is the change in specific mechanical energy (J/kg ≡ m²/s²).

Units

• Inputs: ($v$) in m/s, ($\mathrm{\Delta }v$) in m/s.
• Output: ($\mathrm{\Delta }\epsilon$) in J/kg (m²/s²).

Notes

• This approximation assumes a small, prograde impulse (aligned with velocity) and neglects higher-order terms ($\frac{1}{2}(\mathrm{\Delta }v{)}^2$).
• A more general small-angle form is ($\mathrm{\Delta }\epsilon \approx v\mathrm{\Delta }v\cos \varphi$), where ($\varphi$) is the angle between the velocity vector and the burn direction.

Parameters

Parameter	Type	Description
v	number	Instantaneous speed at burn point (m/s). Must be finite and ≥ 0.
dv	number	Impulsive prograde Δv magnitude (m/s). Must be finite and ≥ 0.

Returns

number

Specific energy gain ($\mathrm{\Delta }\epsilon$) (J/kg).

Throws

If inputs are not numbers.

Throws

If inputs are non-finite or negative.

Example

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

// Earth's GM (μ) in m³/s²
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 ≡ m²/s²)
const deltaEps = oberthEnergyGain(vLEO, dv);

// For intuition, translate Δε to an approximate change in semi-major axis (elliptic case):
// ε = -μ/(2a)  ⇒  Δa ≈ (a² / 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).