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Chapter 3: Problem 294

A spacecraft moving in a west-to-east equatorial orbit is observed by a tracking station located on the equator. If the spacecraft has a perigee altitude \(H=150 \mathrm{km}\) and velocity \(v\) when directly over the station and an apogee altitude of \(1500 \mathrm{km},\) determine an expression for the angular rate \(p\) (relative to the earth) at which the antenna dish must be rotated when the spacecraft is directly overhead. Compute \(p .\) The angular velocity of the earth is \(\omega=0.7292\left(10^{-4}\right) \mathrm{rad} / \mathrm{s}\).

Short Answer

Expert verified
The antenna angular rate is approximately 0.00115 rad/s.

Step by step solution

01

Understand the Problem

We need to find the relative angular rate \( p \) of the tracking station antenna, which accounts for both the spacecraft's movement and Earth's rotation.
02

Calculate the Semi-Major Axis

The perigee \( r_p \) is the Earth's radius plus perigee altitude: \( r_p = R_{e} + H = 6371 \text{ km} + 150 \text{ km} = 6521 \text{ km} \).The apogee \( r_a \) is \( R_{e} + 1500 \text{ km} = 7871 \text{ km} \).The semi-major axis \( a \) is \( a = \frac{r_p + r_a}{2} = \frac{6521 + 7871}{2} = 7196 \text{ km} \).
03

Use the Vis-Viva Equation

The vis-viva equation is \( v^2 = \mu\left(\frac{2}{r} - \frac{1}{a}\right) \), where \( \mu = 398600 \text{ km}^3/\text{s}^2 \), \( r = r_p \).\(v = \sqrt{398600 \left(\frac{2}{6521} - \frac{1}{7196}\right)}\).
04

Compute Orbital Angular Velocity

Using \( v = r_0 \cdot \omega_{orb} \), find \( \omega_{orb} = \frac{v}{r_0} \). Use the \( v \) calculated in the previous step to find \( \omega_{orb} \).
05

Determine Relative Angular Rate for Antenna

The antenna's relative angular speed \( p \) combines spacecraft orbital angular velocity \( \omega_{orb} \) and Earth's angular velocity \( \omega \).\[ p = \omega_{orb} - \omega \]
06

Final Calculation of Angular Rate

Substitute the computed \( \omega_{orb} \) and \( \omega = 0.7292 \times 10^{-4} \text{ rad/s} \) into the equation to find \( p \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spacecraft Motion
Spacecraft motion refers to the movement of a spacecraft as it travels along its orbital path around a celestial body, such as Earth. Understanding spacecraft motion is important for predicting where a spacecraft will be at any given time, which is crucial for successful communication and mission operations.
  • **Orbital Path:** Spacecraft usually follow an elliptical orbit, defined by parameters like altitude, velocity, and gravitational forces.
  • **Tracking Movement:** Monitoring a spacecraft's motion is often carried out by ground-based stations equipped with antennas that track the spacecraft's position. Calculating the rate at which these antennas rotate is essential for maintaining a steady connection with the spacecraft.
As a spacecraft moves in its orbit, it passes directly over specific locations at predictable intervals, which can be calculated using the principles of orbital mechanics.
Angular Velocity
Angular velocity is a measure of how quickly an object rotates around a point or axis. When applied to spacecraft, it describes the speed at which the spacecraft moves along its orbit in terms of the degree or radians it covers per unit time.
  • **Spacecraft's Motion:** The spacecraft's angular velocity helps in determining how fast the spacecraft orbits Earth, which is necessary to calculate the orbital period and relative positions concerning ground stations.
  • **Earth's Rotation:** Earth's own rotation has to be considered when monitoring spacecraft motion. The planet rotates west-to-east, hence affecting the relative angular velocity when observed from any Earth-bound station.
By understanding angular velocity, engineers can predict how long a spacecraft will be within the range of a tracking station. This enables optimal communication windows and navigation adjustments.
Vis-Viva Equation
The Vis-Viva equation is a fundamental formula in orbital mechanics that relates a moving body's speed along a specific trajectory in an orbit. It provides insight into how velocity changes at different points in the orbit.
  • **Equation Format:** The equation is given by: \[ v^2 = \ mu\left(\frac{2}{r} - \frac{1}{a}\right) \] where \( v \) is the velocity, \( r \) is the distance from the center of the body being orbited, \( a \) is the semi-major axis of the orbit, and \( \mu \) is the standard gravitational parameter.
  • **Utility:** By knowing the position (perigee or apogee) and the semi-major axis, the velocity of a spacecraft at any given point can be calculated. This helps to ensure precise orbital adjustments and velocity predictions.
The Vis-Viva equation is a vital tool for mission planning, spacecraft design, and during real-time operations, as it aides in predicting scale velocity variations across an orbital period.
Orbital Angular Rate
The orbital angular rate describes how fast the spacecraft moves along its orbit relative to a fixed point on Earth, such as a tracking antenna. It is a crucial factor when determining the correct angle and speed for antenna rotation.
  • **Calculation Process:** In this context, the orbital angular rate \( \omega_{orb} \) is calculated by considering both the spacecraft's velocity derived from the Vis-Viva equation and the radius at perigee or apogee.
  • **Combining Rates:** To find the relative angular rate \( p \), which is crucial for aligning the tracking antenna, the spacecraft's orbital angular rate is adjusted by subtracting Earth's own rotational angular velocity \( \omega \).
Calculating the orbital angular rate accurately is essential for maintaining a reliable link with the spacecraft as it moves in its orbit, ensuring that the ground-based antenna can effectively track the spacecraft's path even as Earth rotates beneath it.

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