The new position is r = 6867.3(xhat) + 398.8(yhat). The answer to the first problem would be theta = arctan(398.8/6867.3) = 0.058 degrees. But how do you know how much the line of nodes, which is the new position as show in the diagram, should have rotated for a true sun-synchronoous orbit? What is the formula? What are some reasons why there would be any discrepancy between the estimated angle and the actual angle?
The new position is r = 6867.3(xhat) + 398.8(yhat). The answer to the first problem would be theta = arctan(398.8/6867.3) = 0.058 degrees. But how do you know how much the line of nodes, which is the new position as show in the diagram, should have rotated for a true sun-synchronoous orbit? What is the formula? What are some reasons why there would be any discrepancy between the estimated angle and the actual angle?
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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The new position is r = 6867.3(xhat) + 398.8(yhat).
The answer to the first problem would be theta = arctan(398.8/6867.3) = 0.058 degrees.
But how do you know how much the line of nodes, which is the new position as show in the diagram, should have rotated for a true sun-synchronoous orbit? What is the formula?
What are some reasons why there would be any discrepancy between the estimated angle and the actual angle?
![### Explanation of Diagram and Mathematical Concept
The diagram illustrates a coordinate system with the x and y axes. It depicts the original and new positions of a point, marked in blue and orange, respectively.
- **Original Position**: This point is at the x-axis. The description notes this as "original value is x only."
- **New Position**: The point is located at coordinates (new x, new y).
A line is drawn from the original to the new position, forming an angle \( \theta \) with the x-axis. This angle is crucial in determining the change in position.
### Mathematical Expression
The angle \( \theta \) is calculated using the arctangent function:
\[ \Theta = \arctan(y/x) \]
This represents the angle in radians (or degrees) depending on the arctan function used, indicating the slope or rotation from the original x-axis position.
### Questions for Educational Exploration
1. **Give the angle by which the line of nodes has changed**: ______
- Determine how the angle \( \theta \) has shifted from its original alignment.
2. **For a True Sun-Synchronous Orbit**:
- Considering the elapsed time since the propagation started and the specific date in question for part a, calculate how much the line of nodes should have rotated: ______
3. **Discrepancy Analysis**:
- List a couple of reasons why there might be discrepancies between the calculated angles in parts a and b.
This exercise is designed to help students understand rotational dynamics and angular measurements in physics, particularly in orbit analysis for celestial mechanics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2F3a5d6913-f63f-4eb6-a161-606cd2d355e4%2Fy5tvl3_processed.png&w=3840&q=75)
Transcribed Image Text:### Explanation of Diagram and Mathematical Concept
The diagram illustrates a coordinate system with the x and y axes. It depicts the original and new positions of a point, marked in blue and orange, respectively.
- **Original Position**: This point is at the x-axis. The description notes this as "original value is x only."
- **New Position**: The point is located at coordinates (new x, new y).
A line is drawn from the original to the new position, forming an angle \( \theta \) with the x-axis. This angle is crucial in determining the change in position.
### Mathematical Expression
The angle \( \theta \) is calculated using the arctangent function:
\[ \Theta = \arctan(y/x) \]
This represents the angle in radians (or degrees) depending on the arctan function used, indicating the slope or rotation from the original x-axis position.
### Questions for Educational Exploration
1. **Give the angle by which the line of nodes has changed**: ______
- Determine how the angle \( \theta \) has shifted from its original alignment.
2. **For a True Sun-Synchronous Orbit**:
- Considering the elapsed time since the propagation started and the specific date in question for part a, calculate how much the line of nodes should have rotated: ______
3. **Discrepancy Analysis**:
- List a couple of reasons why there might be discrepancies between the calculated angles in parts a and b.
This exercise is designed to help students understand rotational dynamics and angular measurements in physics, particularly in orbit analysis for celestial mechanics.
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