Module 1 ─ Orbit Dynamics I Study Notes

Introduction

• Orbital mechanics extends celestial mechanics to artificial satellites.
• Ideal (Keplerian) vs. real (perturbed) orbits:
  • Perturbations (non-spherical gravity, atmospheric drag, third-body effects, solar radiation pressure, etc.) can hinder or aid orbit control.
• Historical context:
  • Johannes Kepler (1571–1630) supplied empirical laws.
  • Isaac Newton (1643–1727) provided physical derivations via differential equations and universal gravitation.

Developing the Model: Kepler & Newton

• Johannes Kepler
  • Assistant to Tycho Brahe to obtain precise planetary data (esp. Mars).
  • Derived three planetary laws geometrically, no dynamics.
  • Hypothesized a central force toward the Sun.
  • Corrected tidal theory, yet ignored by contemporaries (Galileo, Descartes).
  • Waited ~60 years for Newtonian explanation.
• Isaac Newton
  • Invented differential calculus; modeled nature with equations.
  • Derived Kepler’s laws from $F=Gm<em>1m</em>2/r^2$ and $F=ma$.

Kepler’s Three Laws (Empirical)

1. Law of Orbits
   • Planets travel in ellipses with the Sun at one focus.
   • Ellipse parameters:
     • $a$ – semi-major axis
     • $b$ – semi-minor axis
     • $e$ – eccentricity.
2. Law of Areas (Constant Areal Velocity)
   • Equal areas swept in equal times ⇒ $\dot A = \text{const}$ for a given body.
   • Quantitative form (derivable later): $\dot A = \frac{\mu(1-e^2)}{2h} = \text{const}$.
3. Law of Periods
   • $T^2/a^3 = 4\pi^2/\mu$ ((\mu\equiv G M_{central})).
   • Indicates sweep rate differs between orbits (larger (a) ⇒ longer (T)).

Newton’s Four Fundamental Laws (Applied to Orbital Motion)

1. Inertia: constant velocity (or rest) unless external force acts.
2. Force–Momentum: $\mathbf F = \dfrac{d\mathbf p}{dt},\; \mathbf p=m\mathbf v$; for constant (m): $\mathbf F = m\mathbf a$.
3. Action–Reaction: $\mathbf F<em>{12} = -\mathbf F</em>{21}$.
4. Universal Gravitation: $F = G\dfrac{m<em>1 m</em>2}{r^2}$, with $G = 6.67\times10^{-11}\,\text{m}^3\,\text{kg}^{-1}\,\text{s}^{-2}$.

Work & Energy in Gravitational Fields

• Differential work: $dW = \mathbf F\cdot d\mathbf r$ (only component along motion contributes).
• Total work over path (c): $W<em>{12}=\int</em>{r<em>1}^{r</em>2}!\mathbf F\cdot d\mathbf r$.
• Work–Kinetic-Energy theorem:
  $W<em>{12}=\tfrac12 m v</em>2^2 - \tfrac12 m v_1^2 = \Delta KE.$
• Conservative field ⇒ potential energy (U): $\mathbf F = -\nabla U$.
  • Gravitational potential between two masses (zero at (\infty)):
    $U(r) = -\dfrac{G m<em>1 m</em>2}{r}.$
• Conservation of mechanical energy: $\Delta KE + \Delta PE = 0$ ⇒ $KE+PE = \text{constant}.$

Two-Body Problem (Idealized)

• Only two masses interact by inverse-square gravity; third-body forces neglected.
• Equations in vector form for masses 1 & 2:
  $\mathbf F<em>1 = -G\dfrac{m</em>1 m<em>2}{r^3}\mathbf r,\quad \mathbf F</em>2 = +G\dfrac{m<em>1 m</em>2}{r^3}\mathbf r.$

Angular (Moment of) Momentum

• For a particle of mass (m) in orbit:
  $\mathbf h = \mathbf r\times m\mathbf v$.
• Magnitude (scalar) for planar motion:
  $h = m r v<em>t = m r^2 \dot\theta,$ where (vt) is transverse component.
• In a central force field, $\mathbf h$ is conserved (direction & magnitude).

General Orbit Equation (Conic Section)

• From Newtonian dynamics:
  $r = \frac{h^2/\mu}{1+e\cos(\theta-\theta<em>0)}$ or equivalently $r = \frac{p}{1+e\cos\nu},$ with semilatus rectum $p = h^2/\mu$ and true anomaly (\nu\equiv\theta-\theta0).
• The parameter $e$ determines conic type:
  • $e=0$ ⇒ circle.
  • 0<e<1 ⇒ ellipse.
  • $e=1$ ⇒ parabola.
  • e>1 ⇒ hyperbola.

Analysis of Specific Keplerian Orbits

Circular Orbit (e = 0)

• Radius constant $r=a=p=h^2/\mu$.
• Orbital speed: $v_c = \sqrt{\mu/r}.$
• Period: $T_c = 2\pi\sqrt{r^{3}/\mu}.$

Elliptical Orbit (0 < e < 1)

• Periapsis (closest): (\nu=0^\circ) ⇒
  $r_p = \dfrac{p}{1+e} = a(1-e).$
• Apoapsis (farthest): (\nu=180^\circ) ⇒
  $r_a = \dfrac{p}{1-e} = a(1+e).$
• Velocity at an arbitrary (r):
  $v^2 = \mu\left(\dfrac{2}{r}-\dfrac{1}{a}\right).$
• Period from Kepler III: $T = 2\pi\sqrt{a^{3}/\mu}.$

Parabolic Orbit (e = 1)

• Specific energy = 0.
• Radius relation: $r=\dfrac{p}{1+\cos\nu} = \dfrac{h^2/\mu}{1+\cos\nu}.$
• Periapsis distance: $r_p = p/2.$

Hyperbolic Orbit (e > 1)

• Open trajectory; excess specific energy (>0).
• Approach asymptote defined by eccentricity and semi-major axis (a < 0).

Activity 1 (Practice Problems)

1. Circular Earth orbit with speed $v=7\,\text{km/s}$.
   • Use $v<em>c=\sqrt{\mu</em>E/(R_E+h)}$.
   • Solve for altitude (h). ((\muE = 3.986\times10^{14}\,\text{m}^3\,\text{s}^{-2}); (RE=6378\,\text{km})).
2. Mars elliptical orbit, major axis $2a=40,000\,\text{km}\Rightarrow a=20,000\,\text{km}$.
   • Period: $T=2\pi\sqrt{a^{3}/\mu<em>{Mars}}$, with $\mu</em>{Mars}=4.282\times10^{13}\,\text{m}^3\,\text{s}^{-2}$.
3. Derive velocities:
   • Start from general equation $r=\dfrac{h^2/\mu}{1+e\cos\nu}$ along with energy $v^2=\mu\left(\dfrac{2}{r}-\dfrac{1}{a}\right).$ Specialize to:
     • Circle: set $e=0,\;a=r$ ⇒ $v_c=\sqrt{\mu/r}$.
     • Ellipse: retain $e$, use vis-viva result above.

Reference Constants

• $G = 6.6743\times10^{-11}\,\text{m}^3\,\text{kg}^{-1}\,\text{s}^{-2}$.
• Earth radius: $R_E = 6378\,\text{km}$.
• Earth mass: $M_E = 5.9722\times10^{24}\,\text{kg}$.
• Sun mass: $M_\odot = 1.9891\times10^{30}\,\text{kg}$.
• Mars mass: $M_{Mars}=6.41693\times10^{23}\,\text{kg}$.
• Standard gravitational parameters:
  $\mu<em>E = G M</em>E, \; \mu<em>{Mars}=G M</em>{Mars}.$

Next Topic

• Detailed derivation of the general equation of motion for a body in a Keplerian orbit (start from Newton’s second law in polar coordinates, apply conservation of angular momentum, integrate).