The Motion of Satellites

The Motion of Satellites

Overview of Satellites

  • Definition: Artificial satellites orbiting Earth, such as the International Space Station (ISS) with mass ~4.5 × 10^5 kg and width > 108 m.
  • Importance: Understanding orbits helps analyze planetary motion; Newton’s laws and the law of gravitation are essential tools.

Projectile Motion and Satellite Orbits

  • Initial Concept: A motorcycle rider orbits Earth similarly to how a satellite does when launched at a sufficient speed from a height.
  • Trajectories:
  • Closed Orbits: Trajectories 1-5 are closed orbits, can be elliptic (including circular) paths.
  • Open Orbits: Trajectories 6-7, where the projectile moves away indefinitely without returning.

Circular Orbits

  • Circular Orbital Movement:
  • Simplest and most common type of orbit.
  • Affected solely by gravitational attraction, providing centripetal acceleration.
  • The satellite is not falling directly into Earth but is in a state of continuous free-fall around it.
  • Orbital Equation: For circular motion,
  • Acceleration: a_rad = v^2 / r (where v = orbital speed, r = orbit radius)
  • Gravitational Force: Fg = G * mE * m / r^2

Relationships in Satellite Motion

  • Newton's Second Law Application:
  • Combining gravitational force and centripetal acceleration reveals:
  • G * m_E * m / r^2 = m * v^2 / r
  • Cancelling mass (m) leads to:
    • v^2 = G * m_E / r
    • v = sqrt(G * m_E / r) (13.10)
  • Implications:
  • Orbital speed (v) depends only on the radius of orbit, not the mass of the satellite.

Apparent Weightlessness

  • Astronauts in orbit experience apparent weightlessness, feeling no forces acting on their bodies as they are in synchronous motion with the satellite.
  • This state occurs due to gravity being the only force acting on the satellite and its occupants across various orbit shapes.

Orbital Period and Radius Relationship

  • Deriving Period (T):
  • v = 2πr / T leads to T = 2πr / v (13.11)
  • Relationship noted: Larger orbits have slower speeds and longer periods:
  • T^2 ∝ r^3 (Kepler’s Third Law).

Example Comparisons

  • International Space Station:
  • Radius = 6800 km (400 km above Earth's surface), Speed = 7.7 km/s, Period = 93 min.
  • Moon:
  • Radius = 384,000 km, Speed = 1.0 km/s, Period = 27.3 days.
  • Note: Escape speed is approximately twice the orbital speed for the same radius; for satellites close to any planet, the formula signifies that speed needs to be doubled to escape gravitational pull.

Mechanical Energy in Circular Orbits

  • Total Mechanical Energy (E) in circular orbits:
  • E = K + U = - G * m_E * m / (2r) (13.13)
  • Insights:
  • Energy is negative, confirming that satellites are bound to orbit.
  • Increasing radius increases total mechanical energy, making it less negative.
  • Impact of Atmospheric Drag: Lower orbits might decay due to atmospheric resistance, reducing mechanical energy until the satellite de-orbits.

Applications Beyond Earth

  • Similar gravitational analysis for any object in orbit around a massive body, evidenced by Pluto’s satellites.

Example Calculation Problem

  • Scenario: 1000 kg satellite orbiting 340 km above Earth's surface.
  • Steps to solve include calculating:
    • Radius from altitude
    • Orbital Speed (v) using Eq. 13.10
    • Period (T) using Eq. 13.12
    • Radial Acceleration (a_rad)
    • Work done for placing and escaping orbits.