Orbital Dynamics

The PDF of Yara’s summary can be found here Yara Orbital

1. The n-Body Problem and the Two-Body Problem

The three body Problem

• Involves n gravitationally interacting bodies.

• Equation of motion for the i-th body:

  ${\ddot{r}}_i={\sum }_{j\ne i}^{n}G\frac{m_j}{|{\mathbf{r}}_j-{\mathbf{r}}_i{|}^3}({\mathbf{r}}_j-{\mathbf{r}}_i)$

• This system lacks closed-form solutions due to its complexity, necessitating numerical integration or simplifications like dividing the problem into several two-body problems.

The Two-Body Problem:

• Simplified to two masses interacting gravitationally:

  $r=-\frac{\mu }{r^3}\mathbf{r},\mu =G(m_1+m_2)$

• Leads to conic-section solutions (ellipse, parabola, hyperbola) based on energy and eccentricity (e):

  • e = 0: Circular orbit.
  • 0<e<1: Elliptical orbit.
  • e=1: Parabolic trajectory.
  • e>1: Hyperbolic escape.

• Defines constants of motion:

  • Angular momentum:

    $\mathbf{h}=\mathbf{r}\times \mathbf{v}$

  • Energy:

    $E=\frac{1}{2}{v}^2-\frac{\mu }{r}$

The Restricted Three-Body Problem

• Describes a small body’s motion influenced by two larger bodies (e.g., Earth-Moon or Sun-Earth systems).

• Key features:

  • Lagrange points: Equilibrium points (5 total) where the gravitational forces and centrifugal forces balance.

  • L1, L2, L3: Unstable

  • L4, L5: Stable

  

  • Orbits near Lagrange points, such as Halo orbits, are vital for missions like the James Webb Space Telescope​​.

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2. Orbit Terminology

• Orbits by altitude:

  • Low Earth Orbit (LEO): Altitudes < 2,000 km; used for Earth observation and communication.
  • Geostationary Orbit (GEO): Circular orbit at ~35,786 km, matching Earth’s rotation. Ideal for broadcasting.
  • Lunar Orbit
  • Interstellar Orbit

• Orbits by inclination:

  • prograde i<90°
  • polar i = 90°
  • retrograde i > 90°

• Orbits by function:

  • mission Orbit
  • transfer orbit
  • parking orbit
  • phasing orbit (also some kind of a transfer orbit)

• Molniya Orbit: Highly elliptical orbit with an inclination of ~63.4°, optimized for high-latitude coverage.

  +coverage of high lat, - high cost and radiation

• SSO: Sun Synchronous Orbit

  +constant sun angle, -high cost

• repeating groundtrack: +coverage repeats, -orbit perturbations can resonate

• Orbital Elements: 📚 Memorize

  • a: Semi-major axis (size of the orbit).
  • e: Eccentricity (shape of the orbit).
  • i: Inclination (tilt relative to equatorial plane).
  • Ω: Right ascension of ascending node (RAAN).
  • ω: Argument of perigee.
  • ν: True anomaly (position in orbit).

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3. Orbit Perturbations

• Sources of Perturbations:
  • Imperfect sphere (oblateness of Earth (J2)): This leads to the following effects:
    • nodal regression (ascending node moves) due to a torque which rotates the satellite’s orbit in the equatorial plane)
    • apsidal precession (the argument of periapsis moves) at the critical inclination of 63.435° (Molniya Orbit) this effect cancles out and does not exist.
  • Third-Body Effects: Perturbations due to the Sun, Moon, or other celestial bodies.
  • Atmospheric Drag: Significant in low orbits, reducing altitude and orbital lifetime.
  • Solar Radiation Pressure: Affects large spacecraft in high-altitude orbits​.
• Lagrange Planetary Equations describe how orbital elements evolve under small perturbing forces​.

$$\ddot{r}=-\frac{\mu }{r^3}+f_{pretubation}$$

Excursion: The perturbation forces are modelled either conservative or non-conservative

• conservative: A mathematical formulation for the forces exists (e.g. gravity)
• non-conservative: The forces are not fully calculable (e.g. atmospheric drag)

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4. Orbital Maneuvers and Transfer Orbits

Types of Maneuvers:

• Hohmann Transfer:

  • Most efficient transfer between two circular coplanar orbits.

  • Delta-v (Δv) for each impulse:

    $\Delta v_1=$ $\sqrt{\frac{\mu }{r_1}}\left(\sqrt{\frac{2r_2}{r_1+r_2}}-1\right),\Delta v_2=\sqrt{\frac{\mu }{r_2}}\left(1-\sqrt{\frac{2r_1}{r_1+r_2}}\right)$

  • If a rendez vous is required

$$\beta =\pi -\pi (\frac{1+\frac{r_1}{r_2}}{2}{)}^{\frac{3}{2}}$$

• Bi-Elliptic Transfer: Efficient for large ratio changes in orbit size.
• Plane Changes: Expensive maneuvers that alter inclination​​.

Types of orbital maneuvers:

• Impulsive maneuvers: fire for a short duration fast but propellant-intensive
• Extended maneuvers: often used with ion engines
• Gravity Assist: Uses planetary gravity to alter a spacecraft’s trajectory without propellant. $\Delta v$ is changed without spending propellant Grand tour if multiple planets are used (Cassini used 2 Venus, 1 Earth and 1 Jupiter flybys to get to Saturn)
• Aerobraking: Utilizes atmospheric drag to reduce velocity without propellant.
• Phasing maneuvers: adjustment of the time-position of a spacecraft

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5. Orbit Selection and Design

• Steps:
  1. Establish Orbit Types
  2. Determine Orbit related mission Requirements (also constraints like cost)
  3. Evaluate Orbit performance
  4. Evaluate Orbit Cost
• Specialized orbits like GEO and Sun-synchronous are often selected for unique advantages but entail higher costs​.
• Trade-offs are common, balancing factors like altitude (coverage vs. resolution) and satellite lifetime (drag vs. radiation).

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6. Orbit-Related Mission Requirements

• Earth-Referenced Missions:
  • How to get the Orbit Requirements?
    • What coverage is required?
    • What Sensitivity (e.g. resolution) is requirded?
    • How is the Environment in this Orbit (Radiation, Sunlight for the solar panels)
    • Are the Launch capabilities existing?
    • Is communication possible?
    • How long does the Satellite survive in this orbit?
  • Coverage area, revisit time, and resolution depend on altitude and inclination.
  • Geosynchronous orbits provide continuous coverage, while polar orbits are ideal for global observations​​.
• Space-Referenced Missions:
  • Requirements
    • accessibility
    • reasonable environment
    • good communications
  • Stability and minimal perturbations are key, often requiring high-altitude or Lagrange point orbits.
• Interplanetary Missions:
  • Requirements:
    • launch windows
    • For human spaceflight: “Forgiving” transfer obrits
  • Efficient transfer orbits (e.g., Hohmann or gravity-assist) and robust thermal/radiation shielding​​.

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7. Orbit Cost

• Components:

  • Launch costs: Scaled by payload mass and target orbit.
  • Delta v costs for orbital maneuvers
    • Station keeping and transfer maneuvers
    • Disposal costs: Compliance with debris mitigation guidelines.

• Cost optimization strategies:

  • Use of smaller satellites and reusable launch vehicles.
  • In-orbit refueling​​.

• Delta v Budget:

  • cummulative velocity change for all orbital maneuvers
  • to calculate the propellant mass required
  $$\begin{gathered} \mathbf{Rocket}\mathbf{equation}:\frac{m_0+m_p}{m_0}=exp(\frac{\Delta v}{v_0}) \\ with{v}_0:effectiveexhaustvelocity \end{gathered}$$
  • $\Delta v$ significantly higher than $v_0$ is impossible without staging
  • The Orbital Cost Function (OCF) describes the ratio of mass available in a 185km circular orbit to that available at the mission orbit
    • $OCF=(1+K)exp(\frac{\Delta v}{v_0})-K$
    • with $K\approx 10\%$ which describes the fraction of hardware for the propultion to the propellant mass.

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8. Constellation Design

This part is more of an additional Information because it has been hardly covered in the lecture.

• Design Process:
  • Establish requirements: Coverage, limits on number of satellites, requirements for sensors…
  • Do single satellite orbit trades (except coverage)
  • Do coverage vs Satellite Number trades
  • Look for coverage holes or methods to reduce satellite number
  • Adjust inclination or phasing for maximal intersatellite distances to avoid collision risk
  • Review Rules and document reasons for choices
• Design Principles:
  • Number of satellites, number of orbital planes, and phasing optimized for coverage and reliability.
  • Example: GPS requires 24 satellites in six orbital planes​.
• Principal issues:
  • Coverage, number of satellites, launch options, Environment
• Rules
  • All Satellites at the same Inclination
  • Avoid perigee rotation: If excentric at 63.4 deg
  • Collision avoidance is critical
  • …

Facts to Memorize

1. Keplerian Elements:

   • Orbital Elements:
     • a: Semi-major axis (size of the orbit).
     • e: Eccentricity (shape of the orbit).
     • i: Inclination (tilt relative to equatorial plane).
     • Ω: Right ascension of ascending node (RAAN).
     • ω: Argument of perigee.
     • ν: True anomaly (position in orbit).

   

2. $\Delta v$ Requirements:

   • Key for estimating mission costs. Δv budgets include launch, orbital insertion, station-keeping, and transfer maneuvers​​.
   • Rocket Equation: $\frac{m_0+m_p}{m_0}=exp(\frac{\Delta v}{v_0})$

3. Specialized Orbits:

   • Geostationary (constant ground position), Sun-synchronous (consistent illumination), and Molniya (high latitude coverage)​​.

4. Hohmann Transfer:

   • Most efficient transfer orbit between two circular orbits in the same plane​.

5. Gravity-Assist Maneuver:

   • Uses a planet’s gravity to alter a spacecraft’s trajectory, conserving propellant​.

Formulas

In this part there are too many formulas please check the formula collection from Anna Kubik

Additional Information

Yara Orbital