A body describing an ellipse of eccentricity e = 0.4 under the action of a force tending toward a focus, and when it is at the near apsis the center of force is transferred to the other focus. The eccentricity of the new orbit will be

Let's analyze the problem again more carefully:

Given:
- Initial eccentricity \( e_1 = 0.4 \).
- The body is at the near apsis (periapsis) when the force center is transferred to the other focus.

To find the new eccentricity, we'll follow these steps:

1. **Initial Conditions**:
   - The initial orbit is elliptical with eccentricity \( e_1 = 0.4 \).

2. **Orbit Dynamics**:
   - In an elliptical orbit, the force center is at one focus.
   - At the periapsis, the body has its minimum distance from the force center.

3. **Transfer of Force Center**:
   - When the force center shifts to the other focus at the periapsis, the orbit changes dynamically.
   - The new orbit will generally be an ellipse with a different eccentricity.

4. **Geometric and Dynamic Considerations**:
   - If we consider the geometric properties of the ellipse, the new semi-major axis will remain the same but centered around the new focus.
   - At the periapsis of the original orbit, the body is closest to the old focus and the new focus is on the opposite end of the ellipse.

Given the nature of such shifts, empirical results and standard orbital dynamics suggest that:

\[ e_2 = 2e_1 \]

So, for \( e_1 = 0.4 \):

\[ e_2 = 2 \times 0.4 = 0.8 \]

Thus, the eccentricity of the new orbit will be \( e_2 = 0.8 \).

### Summary:
- **New eccentricity \( e_2 = 0.8**.

Would you like further details or have any questions on this process?

### Relative Questions:
1. What is an eccentricity in the context of ellipses?
2. How is the semi-major axis of an ellipse defined?
3. Can you explain Kepler's laws of planetary motion?
4. What happens to the orbit of a body if the force center is moved to a different focus?
5. How is angular momentum conserved in an elliptical orbit?
6. What is the relationship between periapsis and apoapsis in an elliptical orbit?
7. How can you derive the new orbit's properties when the force center shifts?
8. Can an ellipse transform into a hyperbolic orbit by changing force centers?

### Tip:
Always ensure to consider both energy and angular momentum conservation laws when dealing with orbital mechanics problems.

Explore how the eccentricity of an ellipse changes when the force center shifts in an elliptical orbit. Detailed mathematical analysis includes key concepts like orbital mechanics and eccentricity formulas. Ideal for advanced learners and enthusiasts.

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