Consider a satellite moving in an elliptic orbit about a focus under attraction The semi-major axis and the eccentricity of the orbit are 30000 km and 0.6, respectively. The radial velocity of the satellite at the true anomaly of 60° is (where r is the radius, and µ = 398601 km³/s²).

To calculate the radial velocity of a satellite at a given true anomaly, we need to understand some key concepts about elliptical orbits and apply them to the scenario provided. 

### Key Concepts

1. **True Anomaly (\( \theta \)):** 
   - The angle between the direction of periapsis (closest point) and the current position of the satellite as seen from the main focus of the ellipse.

2. **Semi-major Axis (\( a \)):**
   - Half of the longest diameter of the ellipse.
   - Given as \( a = 30000 \) km.

3. **Eccentricity (\( e \)):**
   - A measure of how much the orbit deviates from being circular.
   - Given as \( e = 0.6 \).

4. **Gravitational Parameter (\( \mu \)):**
   - Product of the gravitational constant and the mass of the central body.
   - Given as \( \mu = 398601 \) km³/s².

5. **Radial Velocity (\( v_r \)):**
   - The component of the velocity vector that points towards or away from the focus of the ellipse.

### Equations Used

1. **Distance from the focus (Radius), \( r \):**

   The distance of the satellite from the focus of the ellipse at a given true anomaly is calculated using:

   \[
   r = \frac{a(1 - e^2)}{1 + e \cos \theta}
   \]

2. **Radial Velocity Equation:**

   The radial velocity component of the satellite is given by:

   \[
   v_r = \sqrt{\frac{\mu}{p}} \cdot e \cdot \sin \theta
   \]

   where:
   - \( p \) is the semi-latus rectum, given by \( p = a(1 - e^2) \).

### Calculation Steps

#### Step 1: Calculate the Semi-latus Rectum (\( p \))

Using the formula for \( p \):

\[
p = a(1 - e^2) = 30000 \cdot (1 - 0.6^2)
\]

\[
p = 30000 \cdot (1 - 0.36) = 30000 \cdot 0.64 = 19200 \text{ km}
\]

#### Step 2: Calculate the Radius (\( r \)) at True Anomaly \( \theta = 60^\circ \)

Using the formula for \( r \):

\[
r = \frac{a(1 - e^2)}{1 + e \cos \theta} = \frac{19200}{1 + 0.6 \cdot \cos 60^\circ}
\]

Given \( \cos 60^\circ = 0.5 \):

\[
r = \frac{19200}{1 + 0.6 \cdot 0.5} = \frac{19200}{1 + 0.3} = \frac{19200}{1.3}
\]

\[
r = 14769.23 \text{ km}
\]

#### Step 3: Calculate the Radial Velocity (\( v_r \))

Using the formula for radial velocity:

\[
v_r = \sqrt{\frac{\mu}{p}} \cdot e \cdot \sin \theta
\]

Substitute the values:

- \(\mu = 398601 \text{ km}^3/\text{s}^2\)
- \(p = 19200 \text{ km}\)
- \(e = 0.6\)
- \(\sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866\)

\[
v_r = \sqrt{\frac{398601}{19200}} \cdot 0.6 \cdot 0.866
\]

\[
v_r = \sqrt{20.76015625} \cdot 0.6 \cdot 0.866
\]

\[
v_r \approx 4.555 \cdot 0.6 \cdot 0.866
\]

\[
v_r \approx 2.366 \cdot 0.866
\]

\[
v_r \approx 2.048 \text{ km/s}
\]

### Conclusion

The radial velocity of the satellite at a true anomaly of 60° is approximately **2.048 km/s**.

---

**Would you like more details or have any questions? Here are some related questions to consider:**

1. What are the differences between radial velocity and transverse velocity in orbital motion?
2. How does eccentricity affect the speed of a satellite in its orbit?
3. How do you calculate the total velocity of a satellite in an elliptical orbit?
4. What role does the gravitational parameter (\( \mu \)) play in orbital mechanics?
5. How do you derive the formula for radial velocity?
6. What is the significance of the semi-major axis in determining orbital characteristics?
7. Can you calculate the angular momentum of a satellite given its velocity and position?
8. How do orbital parameters change if the satellite experiences external forces like atmospheric drag?

**Tip:**  
Understanding how to calculate radial velocity can help predict satellite positions and plan satellite maneuvers, which is crucial in mission planning and satellite operations.

Learn how to calculate the radial velocity of a satellite in an elliptical orbit with a given semi-major axis and eccentricity. Example problem involves a satellite at a true anomaly of 60° with detailed step-by-step solution using orbital mechanics principles. Suitable for undergraduate-level students.

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