The eccentricity \( e \) of an orbiting body can be calculated with the formula:
\[ e = \frac{r_a - r_p}{r_a + r_p} \]
where \( r_p \) is the perihelion distance and \( r_a \) is the aphelion distance.
Given that \( r_p \) is equal to \( 0.45 r_a \), we can denote \( r_a \) as \( r_a \) and \( r_p \) as \( 0.45 r_a \).
Now, let's substitute these values into the eccentricity formula:
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Calculate \( r_a - r_p \): \[ r_a - r_p = r_a - 0.45 r_a = 0.55 r_a \]
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Calculate \( r_a + r_p \): \[ r_a + r_p = r_a + 0.45 r_a = 1.45 r_a \]
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Plug these into the eccentricity formula: \[ e = \frac{0.55 r_a}{1.45 r_a} \]
The \( r_a \) terms cancel out: \[ e = \frac{0.55}{1.45} \]
Now, calculating the value: \[ e \approx \frac{0.55}{1.45} \approx 0.3793 \]
Therefore, the orbital eccentricity of the planet is approximately 0.38.
The answer is 0.38.