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Astrodynamics and orbital mechanics

Derive two-body trajectories, orbital elements, maneuver budgets, and relative-motion scales through twenty worked problems.

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1. Two-body energy and angular momentum

Definitions & inputs. r radius, v speed, μ=G(M+m), ε specific orbital energy, h specific angular momentum, a semimajor axis.

  1. Newton’s central force governs the relative position.

    r¨=−μr/r3\ddot{\mathbf r}=-\mu\mathbf r/r^3

  2. Energy and angular momentum are conserved.

    ϵ=v2/2−μ/r=−μ/(2a),h=r×v\epsilon=v^2/2-\mu/r=-\mu/(2a),\quad\mathbf h=\mathbf r\times\mathbf v

  3. Rearrange conserved energy for vis-viva and use conic geometry.

    v2=μ(2/r−1/a),p=h2/μ=a(1−e2)v^2=\mu(2/r-1/a),\quad p=h^2/\mu=a(1-e^2)

Interpretation. Negative energy indicates a bound ellipse; positive energy indicates a hyperbola.

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2. Orbital geometry and timing

Definitions & inputs. e eccentricity, E eccentric anomaly, M mean anomaly, ν true anomaly, n mean motion.

  1. Periapsis and apoapsis are measured from the central body’s centre.

    rp=a(1−e),ra=a(1+e)r_p=a(1-e),\quad r_a=a(1+e)

  2. Kepler’s third law fixes the orbital period.

    n=μ/a3,T=2π/nn=\sqrt{\mu/a^3},\quad T=2\pi/n

  3. Equal-area motion leads to Kepler’s equation, usually solved numerically for E.

    M=n(t−tp)=E−esin⁡EM=n(t-t_p)=E-e\sin E

Interpretation. Altitude is radius minus the body radius; confusing them produces large errors.

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3. Impulsive transfers and plane changes

Definitions & inputs. r1,r2 are circular-orbit radii; at=(r1+r2)/2; Δi is the angle between orbital planes.

  1. Compare transfer periapsis speed with the initial circular speed.

    Δv1=μ/r1(2r2/(r1+r2)−1)\Delta v_1=\sqrt{\mu/r_1}(\sqrt{2r_2/(r_1+r_2)}-1)

  2. Compare the final circular speed with transfer apoapsis speed for an outward transfer.

    Δv2=μ/r2(1−2r1/(r1+r2))\Delta v_2=\sqrt{\mu/r_2}(1-\sqrt{2r_1/(r_1+r_2)})

  3. Transfer duration is half an ellipse; a pure plane change is the difference between two equal-length velocity vectors.

    ttr=πat3/μ,Δvplane=2vsin⁡(Δi/2)t_{tr}=\pi\sqrt{a_t^3/\mu},\quad\Delta v_{plane}=2v\sin(\Delta i/2)

Interpretation. Combine maneuvers carefully; separately adding ideal burns can overestimate an optimized combined maneuver.

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4. Relative motion and secular perturbations

Definitions & inputs. x radial and y along-track relative coordinates; J2 oblateness coefficient; i inclination.

  1. Linearize central gravity in the rotating circular-orbit frame.

    x¨−2ny˙−3n2x=0,y¨+2nx˙=0\ddot x-2n\dot y-3n^2x=0,\quad\ddot y+2n\dot x=0

  2. Differentiate n∝a⁻³/² for nearby orbits.

    Δn/n≃−32Δa/a\Delta n/n\simeq-\tfrac32\Delta a/a

  3. Oblateness produces a secular node drift; polar orbits have zero leading nodal drift.

    Ω˙=−32J2n(RE/p)2cos⁡i\dot\Omega=-\tfrac32J_2n(R_E/p)^2\cos i

Interpretation. Real orbit determination and mission planning need perturbations, covariance, epochs, and coordinate-frame definitions.

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Graphical worked example

Earth point-mass circular speed. Radius is from Earth’s centre, not altitude. X axis: Circular orbit radius (km). Y axis: Circular speed (km/s).
Earth point-mass circular speed. Radius is from Earth’s centre, not altitude. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Orbit radius from altitude

Definitions & inputs. h=500 km, RE=6371 km.

  1. Choose the governing model and isolate the requested quantity.

    r=RE+hr=R_E+h

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    r=6371+500r=6371+500

  3. Evaluate the expression; the result uses the units shown.

    Result=6871 km\mathrm{Result}=6871\ {\rm km}

Interpretation. Gravity formulas use radius from the centre.

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Example 02. Circular speed at 500 km

Definitions & inputs. r=6871000 m, μ=3.986004418×10¹⁴ SI.

  1. Choose the governing model and isolate the requested quantity.

    vc=μ/rv_c=\sqrt{\mu/r}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    vc=3.986004418×1014/6871000v_c=\sqrt{3.986004418\times10^{14}/6871000}

  3. Evaluate the expression; the result uses the units shown.

    Result=7616.561 m s−1\mathrm{Result}=7616.561\ {\rm m\,s}^{-1}

Interpretation. Speed is relative to an inertial two-body frame.

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Example 03. Circular period

Definitions & inputs. r=6871000 m.

  1. Choose the governing model and isolate the requested quantity.

    T=2πr3/μT=2\pi\sqrt{r^3/\mu}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    T=2π68710003/(3.986004418×1014)T=2\pi\sqrt{6871000^3/(3.986004418\times10^{14})}

  3. Evaluate the expression; the result uses the units shown.

    Result=5668.144 s\mathrm{Result}=5668.144\ {\rm s}

Interpretation. This is about 94.5 minutes.

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Example 04. Specific orbital energy

Definitions & inputs. a=7000000 m.

  1. Choose the governing model and isolate the requested quantity.

    ϵ=−μ/(2a)\epsilon=-\mu/(2a)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ϵ=−3.986004418×1014/(2⋅7000000)\epsilon=-3.986004418\times10^{14}/(2\cdot7000000)

  3. Evaluate the expression; the result uses the units shown.

    Result=−2.847146×107 J kg−1\mathrm{Result}=-2.847146\times10^{7}\ {\rm J\,kg}^{-1}

Interpretation. The negative sign denotes a bound orbit.

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Example 05. Escape speed at radius

Definitions & inputs. r=7000000 m.

  1. Choose the governing model and isolate the requested quantity.

    vesc=2μ/rv_{esc}=\sqrt{2\mu/r}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    vesc=2(3.986004418×1014)/7000000v_{esc}=\sqrt{2(3.986004418\times10^{14})/7000000}

  3. Evaluate the expression; the result uses the units shown.

    Result=10671.73 m s−1\mathrm{Result}=10671.73\ {\rm m\,s}^{-1}

Interpretation. Escape threshold has zero specific energy.

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Example 06. Periapsis radius

Definitions & inputs. a=10000 km, e=0.2.

  1. Choose the governing model and isolate the requested quantity.

    rp=a(1−e)r_p=a(1-e)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    rp=10000(0.8)r_p=10000(0.8)

  3. Evaluate the expression; the result uses the units shown.

    Result=8000 km\mathrm{Result}=8000\ {\rm km}

Interpretation. Subtract Earth radius to obtain altitude.

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Example 07. Apoapsis radius

Definitions & inputs. a=10000 km, e=0.2.

  1. Choose the governing model and isolate the requested quantity.

    ra=a(1+e)r_a=a(1+e)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ra=10000(1.2)r_a=10000(1.2)

  3. Evaluate the expression; the result uses the units shown.

    Result=12000 km\mathrm{Result}=12000\ {\rm km}

Interpretation. Apoapsis is the farthest orbital point.

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Example 08. Periapsis speed

Definitions & inputs. a=10000000 m, rp=8000000 m.

  1. Choose the governing model and isolate the requested quantity.

    vp=μ(2/rp−1/a)v_p=\sqrt{\mu(2/r_p-1/a)}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    vp=μ(2/8000000−1/10000000)v_p=\sqrt{\mu(2/8000000-1/10000000)}

  3. Evaluate the expression; the result uses the units shown.

    Result=7732.404 m s−1\mathrm{Result}=7732.404\ {\rm m\,s}^{-1}

Interpretation. The orbital speed is larger at periapsis.

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Example 09. Eccentricity from apsides

Definitions & inputs. rp=8000 km, ra=12000 km.

  1. Choose the governing model and isolate the requested quantity.

    e=(ra−rp)/(ra+rp)e=(r_a-r_p)/(r_a+r_p)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    e=(12000−8000)/20000e=(12000-8000)/20000

  3. Evaluate the expression; the result uses the units shown.

    Result=0.2 \mathrm{Result}=0.2\

Interpretation. The same distances imply a=10000 km.

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Example 10. Mean motion

Definitions & inputs. a=7000000 m.

  1. Choose the governing model and isolate the requested quantity.

    n=μ/a3n=\sqrt{\mu/a^3}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    n=3.986004418×1014/70000003n=\sqrt{3.986004418\times10^{14}/7000000^3}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.001078008 rad s−1\mathrm{Result}=0.001078008\ {\rm rad\,s}^{-1}

Interpretation. Mean anomaly advances uniformly in this model.

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Example 11. Mean anomaly from eccentric anomaly

Definitions & inputs. E=π/2 rad, e=0.1.

  1. Choose the governing model and isolate the requested quantity.

    M=E−esin⁡EM=E-e\sin E

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    M=π/2−0.1M=\pi/2-0.1

  3. Evaluate the expression; the result uses the units shown.

    Result=1.470796 rad\mathrm{Result}=1.470796\ {\rm rad}

Interpretation. True anomaly and mean anomaly are generally different.

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Example 12. Hohmann first burn

Definitions & inputs. r1=7000 km, r2=14000 km.

  1. Choose the governing model and isolate the requested quantity.

    Δv1=μ/r1(2r2/(r1+r2)−1)\Delta v_1=\sqrt{\mu/r_1}(\sqrt{2r_2/(r_1+r_2)}-1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Δv1=μ/(7×106)(4/3−1)\Delta v_1=\sqrt{\mu/(7\times10^6)}(\sqrt{4/3}-1)

  3. Evaluate the expression; the result uses the units shown.

    Result=1167.379 m s−1\mathrm{Result}=1167.379\ {\rm m\,s}^{-1}

Interpretation. The first tangential burn raises apoapsis.

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Example 13. Hohmann second burn

Definitions & inputs. r1=7000 km, r2=14000 km.

  1. Choose the governing model and isolate the requested quantity.

    Δv2=μ/r2(1−2r1/(r1+r2))\Delta v_2=\sqrt{\mu/r_2}(1-\sqrt{2r_1/(r_1+r_2)})

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Δv2=μ/(14×106)(1−2/3)\Delta v_2=\sqrt{\mu/(14\times10^6)}(1-\sqrt{2/3})

  3. Evaluate the expression; the result uses the units shown.

    Result=979.1496 m s−1\mathrm{Result}=979.1496\ {\rm m\,s}^{-1}

Interpretation. The second burn circularizes at apoapsis.

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Example 14. Hohmann coast time

Definitions & inputs. Transfer a=10500 km.

  1. Choose the governing model and isolate the requested quantity.

    t=πa3/μt=\pi\sqrt{a^3/\mu}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    t=π(10.5×106)3/μt=\pi\sqrt{(10.5\times10^6)^3/\mu}

  3. Evaluate the expression; the result uses the units shown.

    Result=5353.834 s\mathrm{Result}=5353.834\ {\rm s}

Interpretation. Only half the transfer ellipse is traversed.

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Example 15. Ten-degree plane change

Definitions & inputs. v=7500 m/s; Δi=10°.

  1. Choose the governing model and isolate the requested quantity.

    Δv=2vsin⁡(Δi/2)\Delta v=2v\sin(\Delta i/2)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Δv=15000sin⁡5∘\Delta v=15000\sin5^\circ

  3. Evaluate the expression; the result uses the units shown.

    Result=1307.336 m s−1\mathrm{Result}=1307.336\ {\rm m\,s}^{-1}

Interpretation. Plane changes are expensive at high speed.

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Example 16. Synchronous orbit radius

Definitions & inputs. Desired inertial period T=86164 s.

  1. Choose the governing model and isolate the requested quantity.

    r=[μ(T/2π)2]1/3r=[\mu(T/2\pi)^2]^{1/3}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    r=[μ(86164/2π)2]1/3r=[\mu(86164/2\pi)^2]^{1/3}

  3. Evaluate the expression; the result uses the units shown.

    Result=4.216414×107 m\mathrm{Result}=4.216414\times10^{7}\ {\rm m}

Interpretation. Geostationary also requires an equatorial, prograde, circular orbit.

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Example 17. Hyperbolic excess speed

Definitions & inputs. Specific energy ε=4.5×10⁶ J/kg.

  1. Choose the governing model and isolate the requested quantity.

    v∞=2ϵv_\infty=\sqrt{2\epsilon}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    v∞=9×106v_\infty=\sqrt{9\times10^6}

  3. Evaluate the expression; the result uses the units shown.

    Result=3000 m s−1\mathrm{Result}=3000\ {\rm m\,s}^{-1}

Interpretation. This is asymptotic relative speed to the central body.

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Example 18. Specific angular momentum

Definitions & inputs. Circular r=7000000 m.

  1. Choose the governing model and isolate the requested quantity.

    h=μrh=\sqrt{\mu r}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    h=(3.986004418×1014)(7000000)h=\sqrt{(3.986004418\times10^{14})(7000000)}

  3. Evaluate the expression; the result uses the units shown.

    Result=5.282237×1010 m2 s−1\mathrm{Result}=5.282237\times10^{10}\ {\rm m^2\,s}^{-1}

Interpretation. This is angular momentum per unit orbiting mass.

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Example 19. Nearby-orbit drift rate

Definitions & inputs. a=7000 km, Δa=1 km.

  1. Choose the governing model and isolate the requested quantity.

    Δn≃−32nΔa/a\Delta n\simeq-\tfrac32n\Delta a/a

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Δn=−32μ/(7×106)3(1/7000)\Delta n=-\tfrac32\sqrt{\mu/(7\times10^6)^3}(1/7000)

  3. Evaluate the expression; the result uses the units shown.

    Result=−2.310016×10−7 rad s−1\mathrm{Result}=-2.310016\times10^{-7}\ {\rm rad\,s}^{-1}

Interpretation. The higher orbit advances more slowly in mean anomaly.

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Example 20. Polar J2 nodal drift

Definitions & inputs. Inclination i=90°.

  1. Choose the governing model and isolate the requested quantity.

    Ω˙=−32J2n(RE/p)2cos⁡i\dot\Omega=-\tfrac32J_2n(R_E/p)^2\cos i

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    cos⁡90∘=0\cos90^\circ=0

  3. Evaluate the expression; the result uses the units shown.

    Result=0 rad s−1\mathrm{Result}=0\ {\rm rad\,s}^{-1}

Interpretation. Other perturbations may still move the node.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.