Derivation for Aphelion and Perihelion Distances

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In summary, the formulas for perihelion and aphelion distances in an orbit with SMA a and eccentricity e are a(1-e) and a(1+e), respectively. This follows from the definition of eccentricity and the fact that the sum of the distances to the focal points is always equal to 2a for any point on the ellipse.
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bruhh
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I know the formulas for perihelion and aphelion distances in an orbit with SMA a and eccentricity e are a(1-e) and a(1+e), respectively. However, why is this? I can't seem to find any derivations for this anywhere and also cannot figure this out myself.
 
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Not sure what you mean by "why" here. The focus-directrix definition of an ellipse gives you this more or less instantly, but you may have to work harder with other definitions. Or are you trying to prove it from ##F=GMm/r^2##?

Basically, you have told us where you want to go but not where you are. That makes it tricky to give directions.
 
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See any classical mechanics textbook
 
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bruhh said:
I know the formulas for perihelion and aphelion distances in an orbit with SMA a and eccentricity e are a(1-e) and a(1+e), respectively. However, why is this? I can't seem to find any derivations for this anywhere and also cannot figure this out myself.
It follows from the definition of the eccentricity: the focal points of an ellipse are at a distance ## \pm e a ## from the center. And for perihelion and aphelion, as for any points on the ellipse, the sum of the distances to the foci must be ## 2a ##.
 

Related to Derivation for Aphelion and Perihelion Distances

What is the definition of aphelion and perihelion distances?

Aphelion and perihelion distances refer to the maximum and minimum distances, respectively, between a celestial object (such as a planet) and the Sun in its orbit around the Sun. These distances are measured from the center of the Sun to the closest and farthest points of the object's orbit.

How are aphelion and perihelion distances calculated?

The aphelion and perihelion distances can be calculated using the equation a(1±e), where a is the semi-major axis of the object's orbit and e is the eccentricity of the orbit. The semi-major axis is the average distance between the object and the Sun, while the eccentricity represents the shape of the orbit.

What factors influence the aphelion and perihelion distances of a celestial object?

The main factor that influences the aphelion and perihelion distances of a celestial object is its eccentricity. Objects with higher eccentricity have larger differences between their aphelion and perihelion distances. Other factors that can affect these distances include the gravitational pull of other celestial objects and any changes in the object's orbit over time.

Why are aphelion and perihelion distances important in astronomy?

Aphelion and perihelion distances are important in astronomy because they help us understand the orbital characteristics of celestial objects. By studying these distances, we can determine the shape and size of an object's orbit, as well as its speed and direction of motion. This information is crucial for understanding the dynamics of our solar system and the universe.

How do aphelion and perihelion distances affect the climate on Earth?

The Earth's distance from the Sun at different points in its orbit (aphelion and perihelion) does have a small impact on the climate. However, the tilt of the Earth's axis and its rotation around the Sun are the main factors that determine our seasons and climate. The Earth's elliptical orbit does not significantly affect the amount of solar radiation received by the planet.

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