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.