6.5 Q-9

Question Statement

The moon orbits the Earth in an elliptical path with Earth at one focus. The major and minor axes of the moon’s orbit are given as $786,806 , \text{km}$ and $767,746 , \text{km}$, respectively. Determine the apogee (greatest distance) and perigee (least distance) of the moon from Earth.

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Background and Explanation

In an elliptical orbit:

• The major axis corresponds to the longest diameter of the ellipse, and half of it ($a$) is the semi-major axis.
• The minor axis corresponds to the shortest diameter, and half of it ($b$) is the semi-minor axis.
• The distance from the center to the focus ($c$) is related to the eccentricity ($e$) by $c = a e$.
• The apogee (farthest distance) is calculated as $a + c$, while the perigee (nearest distance) is $a - c$.

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Solution

Step 1: Identify given data

• Total major axis $2a = 786,806 , \text{km}$ ⟹ $a = \frac{786,806}{2} = 384,403 , \text{km}$.
• Total minor axis $2b = 767,746 , \text{km}$ ⟹ $b = \frac{767,746}{2} = 383,873 , \text{km}$.

Step 2: Calculate $c$, the distance from the center to the focus

From the ellipse equation, $c^2 = a^2 - b^2$:

1. Calculate $a^2$:

$$a^2 = 384,403^2 = 147,765,666,409 , \text{km}^2.$$

2. Calculate $b^2$:

$$b^2 = 383,873^2 = 147,358,480,129 , \text{km}^2.$$

3. Compute $c^2$:

$$c^2 = a^2 - b^2 = 147,765,666,409 - 147,358,480,129 = 407,186,280 , \text{km}^2.$$

4. Solve for $c$:

$$c = \sqrt{407,186,280} \approx 20,178.86 , \text{km}.$$

Step 3: Calculate the apogee and perigee

1. Apogee (greatest distance):

$$\text{Apogee} = a + c = 384,403 + 20,178.86 \approx 404,582 , \text{km}.$$

2. Perigee (least distance):

$$\text{Perigee} = a - c = 384,403 - 20,178.86 \approx 364,224 , \text{km}.$$

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Key Formulas or Methods Used

1. Semi-Major and Semi-Minor Axes:

$$a = \frac{\text{Major Axis}}{2}, \quad b = \frac{\text{Minor Axis}}{2}.$$

2. Distance from Center to Focus:

$$c^2 = a^2 - b^2.$$

3. Apogee and Perigee:

$$\text{Apogee} = a + c, \quad \text{Perigee} = a - c.$$

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Summary of Steps

1. Extract data: $a = 384,403 , \text{km}, b = 383,873 , \text{km}$.
2. Compute $c^2 = a^2 - b^2$, giving $c = 20,178.86 , \text{km}$.
3. Calculate apogee $a + c = 404,582 , \text{km}$.
4. Calculate perigee $a - c = 364,224 , \text{km}$.

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Final Answer

• Apogee: $\mathbf{404,582 , \text{km}}$.
• Perigee: $\mathbf{364,224 , \text{km}}$.