At engine cutoff, we compute osculating orbital elements to evaluate mission success.

The Two-Body Problem¶

After leaving the atmosphere, the rocket follows a Keplerian orbit around Earth.

Assumptions¶

1. Point masses: Earth and spacecraft treated as point masses
2. No perturbations: No atmospheric drag, solar radiation, or third-body effects
3. Spherical Earth: Gravitational field is \(\mu/r^2\)

Governing Equation¶

Where \(\mu = GM_\oplus = 3.986 \times 10^{14}\) m³/s².

Step 1: Specific Orbital Energy¶

Derivation¶

The total mechanical energy per unit mass is conserved:

Step 1.1: Kinetic energy per unit mass = \(\frac{1}{2}v^2\)

Step 1.2: Potential energy per unit mass = \(-\frac{\mu}{r}\) (zero at infinity)

Step 1.3: Sum gives specific orbital energy:

Physical Interpretation¶

Implementation:

# [Eq. 18] Specific orbital energy
eps = 0.5 * v * v - mu / r

Step 2: Specific Angular Momentum¶

Derivation¶

Angular momentum per unit mass is also conserved:

For planar motion:

Where \(v_\perp = v\cos\gamma\) is the tangential velocity component.

Physical Interpretation¶

• \(h = 0\): Radial motion (impact trajectory)
• Maximum \(h\) for given energy: Circular orbit
• \(\gamma = 0\): All velocity is tangential (\(h = rv\))

Implementation:

# [Eq. 19] Specific angular momentum
h_ang = r * v * jnp.cos(gamma)

Step 3: Eccentricity¶

Derivation¶

The eccentricity relates energy and angular momentum:

Step 3.1: Start from the vis-viva equation:

Step 3.2: For an ellipse:

Step 3.3: The semi-latus rectum is:

Step 3.4: Eccentricity from \(p = a(1 - e^2)\):

Numerical Guard¶

To handle numerical errors that might give \(e^2 < 0\):

Implementation:

# [Eq. 20] Eccentricity with numerical guard
e = jnp.sqrt(jnp.maximum(0.0, 1.0 + (2.0 * eps * h_ang * h_ang) / (mu * mu)))

Eccentricity Values¶

Step 4: Semi-Major Axis¶

Derivation¶

From the energy equation:

Solving for \(a\):

Implementation:

# [Eq. 21] Semi-major axis (valid for bound orbits)
a = -mu / (2.0 * eps)

Step 5: Apoapsis and Periapsis¶

Derivation¶

For an ellipse with semi-major axis \(a\) and eccentricity \(e\):

Altitudes above Earth's surface:

Implementation:

r_apo = a * (1.0 + e)
r_peri = a * (1.0 - e)

Mission Success Criteria¶

For a circular orbit at target altitude \(h_{\text{target}}\):

Target State¶

Success Metrics¶

Complete Implementation¶

def orbit_diagnostics(*, y: Array, earth: EarthParams):
    """Two-body orbit diagnostics at cutoff per LaTeX Report (Section 7).

    Equations:
      - Specific Energy (epsilon): [Eq. 18]
      - Specific Angular Momentum (h): [Eq. 19]
      - Eccentricity (e): [Eq. 20]
      - Semi-major Axis (a): [Eq. 21]
    """
    r = y[_R]
    v = y[_V]
    gamma = y[_GAMMA]

    mu = earth.mu

    # [Eq. 18] Specific energy
    eps = 0.5 * v * v - mu / r

    # [Eq. 19] Specific angular momentum
    h_ang = r * v * jnp.cos(gamma)

    # [Eq. 20] Eccentricity
    e = jnp.sqrt(jnp.maximum(0.0, 1.0 + (2.0 * eps * h_ang * h_ang) / (mu * mu)))

    # [Eq. 21] Semi-major axis
    a = -mu / (2.0 * eps)

    # Apoapsis and periapsis
    r_apo = a * (1.0 + e)
    r_peri = a * (1.0 - e)

    return eps, h_ang, a, e, r_apo, r_peri

Orbital Parameters in Output¶

The simulation returns orbital diagnostics in the JSON output:

{
  "trajectory": {
    "orbit": {
      "specific_energy": -2.94e7,
      "specific_angular_momentum": 5.13e10,
      "semi_major_axis": 6578137,
      "eccentricity": 0.000498,
      "apoapsis": 6581414,
      "periapsis": 6574860
    }
  }
}

References¶

1. Battin, R.H. (1999). An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series. Chapters 3-4.

2. Curtis, H.D. (2013). Orbital Mechanics for Engineering Students. Butterworth-Heinemann. Chapter 2 - "The Two-Body Problem".

3. Vallado, D.A. (2013). Fundamentals of Astrodynamics and Applications. Microcosm Press. Chapter 2.

Next Steps¶

• Flight Phases - Hybrid system dynamics
• Shooting Method - How we achieve circular orbits