Engineering Math - Orbital Motion  

 

 

 

Orbital Motion

This note concentrates on the mathematical explanations and derivations presented in the transcript on orbital mechanics. It covers Newton’s law of gravity, the need for horizontal velocity to achieve orbit, derivations of orbital velocity and period, application to a real-world problem, and additional insights regarding trajectory shapes and weightlessness.

Newton’s Law of Gravity and the Need for Horizontal Velocity

  • Concept Introduction: The  rockets do not simply go “straight up” into space. Instead, to achieve orbit, a spacecraft must build a significant horizontal (tangential) velocity so that while gravity pulls it toward Earth, its path curves away from the surface.
  • Thought Experiment: Using the example of throwing a rock from a tall mountain, it is shown that with a low horizontal speed the rock follows a parabolic path and eventually hits the ground. Increasing the horizontal speed leads to trajectories that nearly “miss” the ground, eventually resulting in a complete orbit when the horizontal speed is high enough.

Trajectory Shapes

The shape of an orbit is determined by the initial velocity of an object relative to the central gravitational force. These shapes—parabola, ellipse, circle, and hyperbola—are mathematical solutions to orbital motion under gravitational influence. Below is a detailed explanation of each type of orbit, how they arise, and their unique characteristics.

Parabolic Trajectories

  • How They Occur: - When an object is launched with an initial velocity that is insufficient to maintain a stable orbit, it follows a parabolic path. - Gravity pulls the object back to the central body before it completes a full revolution.
  • Characteristics: - The object rises along a curved path, slows as it reaches its peak, and then accelerates back toward the surface, intersecting with the ground. - The trajectory is symmetrical and governed by the laws of projectile motion.
  • Examples: - A stone thrown horizontally or upward from a hill. - Sub-orbital rockets that briefly leave the atmosphere but do not achieve orbital speed.

Elliptical Orbits

Elliptical orbits are the most common type of orbit for natural celestial bodies and many artificial satellites. They occur when the object’s initial velocity is sufficient to sustain a closed orbit but does not match the speed needed for a perfect circular path.

  • Two Key Scenarios:
    • Near-Earth Elliptical Orbit:
      • How It Happens: The object’s speed is only slightly above the threshold required to avoid hitting the ground.
      • Key Characteristics: - Speed is highest at perigee due to the stronger gravitational pull. - Speed is lowest at apogee, where gravity’s pull weakens.
      • Practical Examples: Satellites in low Earth orbit (LEO) often follow near-Earth ellipses, with higher speeds near perigee to maintain their trajectory.
    • Farther Elliptical Orbit:
      • How It Happens: The object is launched with more energy, resulting in an orbit with a much higher apogee (greater eccentricity).
      • Key Characteristics: - The difference between apogee and perigee becomes more pronounced. - Requires greater horizontal speed than a near-Earth ellipse.
      • Energy Distribution: - At perigee: Kinetic energy is maximized, and potential energy is minimized. - At apogee: Potential energy is maximized, and kinetic energy is minimized.

Circular Orbit

  • How It Happens: Achieved when the object’s horizontal velocity is exactly equal to the velocity required to balance gravitational pull at a given altitude.
  • Key Characteristics: - Speed is constant throughout the orbit. - The orbit’s radius is fixed, and the shape is perfectly symmetrical.
  • Practical Applications: Common for geostationary satellites, which maintain a fixed position relative to a point on Earth’s surface.

Hyperbolic Trajectories

  • How It Happens: The object gains sufficient kinetic energy to overcome the gravitational potential energy of the central body completely.
  • Key Characteristics: - The trajectory does not form a closed loop. - As the object moves away, its velocity gradually decreases but never reaches zero.
  • Practical Examples: - Spacecraft leaving Earth for missions to other planets. - Comets or asteroids passing near Earth on paths that take them back into deep space.

Visualizing the Transition Between Orbits

  • Low Speed: Leads to parabolic paths that intersect with the surface.
  • Moderate Speed: Results in elliptical orbits, with the eccentricity increasing as speed increases.
  • Exact Speed for a Circular Orbit: Achieves a perfectly symmetrical orbit with a fixed radius.
  • High Speed (Above Escape Velocity): Leads to hyperbolic trajectories that escape the gravitational influence of the planet.

Key Insights

  • Energy’s Role: The balance between kinetic and potential energy determines the shape and stability of the orbit.
  • Speed and Shape: The faster the object’s horizontal speed, the less bound the orbit becomes, transitioning from circular to elliptical to hyperbolic.
  • Applications: Each orbit type serves unique purposes in science and exploration, from launching satellites to sending spacecraft to distant planets.

Deriving the Orbital Velocity

In Circular Orbit

Newton’s Gravitational Force

The gravitational force between two bodies is given by

F = (G M m) / R2,

where G is the gravitational constant, M is the mass of the central body (e.g., Earth), m is the mass of the orbiting object, and R is the distance from the center of the planet to the object.

Centripetal Force Requirement

For an object in circular motion, the required centripetal acceleration is

ac = v2 / R,

so the centripetal force needed is

Fc = m (v2 / R).

Equating Forces

Setting the gravitational force equal to the centripetal force gives:

(G M m) / R2 = m (v2 / R).

Canceling the mass m results in

v2 = (G M) / R,

so the orbital velocity is:

v = sqrt(G M / R).

Key Point: The orbital speed depends only on the mass of the planet and the orbital radius; it does not depend on the mass of the orbiting object.

Deriving the Orbital Period

In Circular Orbit

Definition

The orbital period T is the time it takes to complete one full orbit. For a circular orbit, the distance traveled is the circumference, which is 2πR.

Relation to Velocity

The period is given by T = (2πR) / v.

Substitution of Orbital Velocity

Substituting v = sqrt(G M / R) into the period equation yields:

T = 2πR / sqrt((G M)/R)

this can be rewriten as follows

T = 2πR (1/ sqrt((G M/R)))

this can be rewritten as

T = 2πR (sqrt(1)/ sqrt((G M/R)))

this can be rewritten as

T = 2πR (sqrt(1/(G M/R))

this can be rewritten as

T = 2πR (sqrt(R/(G M))

this can be rewritten as

T = 2πR (R1/2/sqrt(G M))

which simplifies to:

T = 2π R3/2 sqrt( (G M)).

Interpretation: This formula shows that the orbital period increases with the orbital radius (specifically with the 3/2 power of R) and decreases with the square root of the planet's mass.

Application to a Real-World Problem

Example 1 >

  • Example Calculation: The transcript provides an example where a satellite orbits 780 km above the Earth’s surface.
  • Determining Orbital Radius: R = (Earth’s radius) + (altitude). With Earth’s radius ≈ 6.38×106 m and altitude = 780 km = 7.80×105 m, we get: R ≈ 6.38×106 m + 7.80×105 m = 7.16×106 m.
  • Calculating Orbital Velocity: Using the formula v = sqrt(G M / R) with: G = 6.67×10-11 N·m2/kg2 and M = 5.97×1024 kg, the calculation becomes: v = sqrt((6.67×10-11 × 5.97×1024) / 7.16×106), resulting in approximately 7.46×103 m/s (or 7.46 km/s).

YouTube