The eccentricity of the earth's orbit is `0.0167`, the ratio of its maximum speed in its orbit to its minimum speed is

Options

A. `2.507`
B. `1.0339`
C. `8.324`
D. `1.000`

Answer

To find the ratio of the maximum speed to the minimum speed of the Earth in its orbit, we can use the relationship derived from the conservation of angular momentum. The steps are as follows:

### Step 1: Understand the Concept of Eccentricity
The eccentricity (e) of an orbit is a measure of how much it deviates from being circular. For Earth's orbit, the eccentricity is given as \( e = 0.0167 \).

### Step 2: Identify the Positions in the Orbit
In an elliptical orbit, there are two key positions:
- **Perigee**: The point where the Earth is closest to the Sun (maximum speed).
- **Apogee**: The point where the Earth is farthest from the Sun (minimum speed).

### Step 3: Use the Angular Momentum Conservation
The angular momentum (L) of the Earth in its orbit can be expressed as:
\[ L = m \cdot v \cdot r \]
where:
- \( m \) is the mass of the Earth,
- \( v \) is the orbital speed,
- \( r \) is the distance from the Sun.

At perigee (closest approach):
\[ L_{perigee} = m \cdot v_{max} \cdot (a - ae) \]
At apogee (farthest point):
\[ L_{apogee} = m \cdot v_{min} \cdot (a + ae) \]

### Step 4: Set the Angular Momenta Equal
Since there are no external torques acting on the Earth, we can equate the angular momenta:
\[ m \cdot v_{max} \cdot (a - ae) = m \cdot v_{min} \cdot (a + ae) \]

### Step 5: Cancel Out Common Terms
We can cancel \( m \) from both sides:
\[ v_{max} \cdot (a - ae) = v_{min} \cdot (a + ae) \]

### Step 6: Rearrange to Find the Speed Ratio
Rearranging gives us:
\[ \frac{v_{max}}{v_{min}} = \frac{(a + ae)}{(a - ae)} \]

### Step 7: Substitute the Eccentricity
Using the eccentricity \( e \):
\[ \frac{v_{max}}{v_{min}} = \frac{(a(1 + e))}{(a(1 - e))} \]
This simplifies to:
\[ \frac{v_{max}}{v_{min}} = \frac{1 + e}{1 - e} \]

### Step 8: Plug in the Value of Eccentricity
Substituting \( e = 0.0167 \):
\[ \frac{v_{max}}{v_{min}} = \frac{1 + 0.0167}{1 - 0.0167} \]
\[ = \frac{1.0167}{0.9833} \]

### Step 9: Calculate the Ratio
Calculating the above expression:
\[ \frac{v_{max}}{v_{min}} \approx 1.0339 \]

### Final Answer
Thus, the ratio of the maximum speed to the minimum speed of the Earth in its orbit is approximately **1.0339**.

---