A satellite moving around the earth in an elliptical orbit of eccentricity of 0.5. For the flight path angle 20 deg the value of eccentric anomaly is

To solve the problem, we need to use the relationship between the flight path angle (\(\gamma\)), the eccentricity (\(e\)), and the eccentric anomaly (\(E\)) in an elliptical orbit.

### Relationship:
The flight path angle \(\gamma\) is related to the eccentric anomaly \(E\) by the following equation:
\[
\tan(\gamma) = \frac{e \sin(E)}{1 - e \cos(E)}
\]
Where:
- \( \gamma = 20^\circ \)
- \( e = 0.5 \)

### Step-by-Step Solution:

1. **Convert the flight path angle to radians**:
   \[
   \gamma = 20^\circ = \frac{20 \times \pi}{180} = \frac{\pi}{9} \, \text{radians}
   \]

2. **Set up the equation**:
   \[
   \tan\left(\frac{\pi}{9}\right) = \frac{0.5 \sin(E)}{1 - 0.5 \cos(E)}
   \]
   Let \( t = \tan\left(\frac{\pi}{9}\right) \).

3. **Simplify and solve for \(E\)**:
   \[
   t = \frac{0.5 \sin(E)}{1 - 0.5 \cos(E)}
   \]
   \[
   t \left(1 - 0.5 \cos(E)\right) = 0.5 \sin(E)
   \]
   \[
   t - 0.5 t \cos(E) = 0.5 \sin(E)
   \]
   \[
   t = 0.5 \sin(E) + 0.5 t \cos(E)
   \]
   \[
   2t = \sin(E) + t \cos(E)
   \]
   \[
   \sin(E) = 2t - t \cos(E)
   \]

This is a transcendental equation, meaning it doesn’t have an algebraic solution, so numerical methods or approximation are usually employed.

Let's compute the value numerically to find \(E\).The eccentric anomaly \(E\) corresponding to the flight path angle of 20 degrees and an eccentricity of 0.5 is approximately **23.16 degrees**.

Would you like any more details or have any further questions? Here are some related questions you might find interesting:

1. What is the relationship between eccentric anomaly and true anomaly?
2. How does the orbital eccentricity affect the shape of the orbit?
3. Can you derive the formula used to relate the flight path angle with eccentric anomaly?
4. How do you calculate the true anomaly from the eccentric anomaly?
5. What is the significance of the flight path angle in satellite orbits?

**Tip:** When dealing with orbital mechanics, always check if angles are in radians or degrees, as this can affect the accuracy of your calculations.

Learn how to calculate the eccentric anomaly (E) for an elliptical orbit with a flight path angle of 20 degrees and an eccentricity of 0.5. Detailed solution using trigonometric equations and orbital mechanics concepts.

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