Documentation / @interstellar-tools/equations / j2ArgumentOfPerigeeRate

Function: j2ArgumentOfPerigeeRate()

function j2ArgumentOfPerigeeRate(
  J2: number,
  n: number,
  Re: number,
  a: number,
  i: Radians,
  e: number
): number;

Defined in: orbits/j2-argument-of-perigee-rate.ts:55

Compute argument of perigee rotation (J2 secular drift of ω) for an orbit about an oblate body.

Equation (common approximation):

$$\dot{\omega }\approx \frac{3}{4}{J}_{2}n{\left(\frac{R_e}{a}\right)}^2\frac{(5{\cos }^{2}i-1)}{(1-e^2{)}^2}$$

Where:

• $J_2$ is the second zonal harmonic (dimensionless)
• $n$ is mean motion (rad/s)
• $R_e$ is the body's equatorial radius (m or km)
• $a$ is semi-major axis (m or km)
• $i$ is inclination (radians)
• $e$ is eccentricity (dimensionless)

Units:

• If n is in rad/s and i is in radians, the result is in rad/s.
• Re and a must use the same distance unit.

Parameters

Parameter	Type	Description
J2	number	Second zonal harmonic (finite).
n	number	Mean motion (finite, >= 0), in rad/s.
Re	number	Equatorial radius (finite, > 0).
a	number	Semi-major axis (finite, > 0).
i	Radians	Inclination (finite), in radians.
e	number	Eccentricity (finite, 0 <= e < 1).

Returns

number

Argument of perigee drift rate $\dot{\omega }$ in rad/s.

Throws

If any input is not finite.

Throws

If n < 0, Re <= 0, a <= 0, or e is outside [0, 1].

Example

// Earth-ish example (units must be consistent):
const J2 = 1.08262668e-3;
const n = 0.001078; // rad/s (roughly LEO)
const Re = 6378.137; // km
const a = 7000; // km
const i = ((63.4 * Math.PI) / 180) as Radians; // "critical inclination" region
const e = 0.001;

const omegaDot = j2ArgumentOfPerigeeRate(J2, n, Re, a, i, e);

See

• https://science.nasa.gov/wp-content/uploads/2023/05/GDC_OrbitPrimer.pdf NASA GSFC - GDC Orbit Primer (shows apsidal precession dω/dt from J2; equivalent to the (5cos²i−1) form)
• https://en.wikipedia.org/wiki/Apsidal_precession Wikipedia - Apsidal precession (background + links on causes incl. quadrupole/oblateness)
• https://freeflyer.com/\_freeflyeruniversityguide/j2_perturbation.htm FreeFlyer University Guide - J2 perturbation (mission-design oriented discussion of RAAN & argument of perigee drift)
• https://juliaspace.github.io/SatelliteToolboxPropagators.jl/stable/man/propagators/j2/ SatelliteToolboxPropagators.jl - J2 analytical propagator (lists secular ω(t) rate with the (4−5sin²i) term)
• https://articles.adsabs.harvard.edu/pdf/1959AJ.....64..367K Kozai (1959) - The Motion of a Close Earth Satellite (classic primary source on secular J2 effects)
• https://help.agi.com/stk/Content/stk/vehSat_orbitProp_2bodyJ2J4.htm AGI/STK Help - Two-Body + J2/J4 perturbations (notes secular drift in argument of perigee)