Kepler's First Law Calculator

This calculator computes the orbital distance of planets from the Sun using Kepler's First Law of Planetary Motion. Kepler's laws are fundamental to the understanding of astronomy and physics, describing the motion of planets around the Sun. The first law, in particular, reveals that the orbits of planets are ellipses, with the Sun at one of the two foci, a significant advancement from the circular orbits proposed in earlier models.
CalculationInputs﻿﻿p
:1.496e+11m​
﻿
﻿﻿θ
:200.00deg​
﻿
﻿﻿e
:0.0167​
﻿
Output﻿﻿r
:1.519e+11m​
﻿
﻿﻿r=1+e cos(θ)p​​
﻿
Where:
﻿﻿p
 is the semi-latus rectum, a measure of how 'wide' the ellipse is. It is the distance from the focus 1 to the orbit perpendicular to the x-axis, measured in meters (﻿m
) where ﻿p=ab2​
.
﻿﻿e
 is the eccentricity of the ellipse, a dimensionless measure of how much the orbit deviates from a perfect circle, where ﻿e=1−a2b2​​
 for ﻿0≤e≤1
.
﻿﻿θ
 is the angle to the planet's current position from its closest approach (periapsis), as seen from the Sun, measured in degrees (﻿deg
).
﻿﻿r
 is the distance from the Sun to the planet, which varies as the planet orbits the Sun, measured in meters (﻿m
).
ExplanationKepler’s First Law, often termed the Law of Ellipses, asserts that planets orbit the Sun along an elliptical path, with the Sun occupying one of the ellipse's two foci. Unlike a circle, which has a single central point, an ellipse is stretched and contains two focal points. The sum of the distances from the ellipse's edge to these two foci remains constant for any point on the ellipse.
﻿
Diagram illustrating Kepler’s First Law
In celestial mechanics, the elliptical orbit has two notable points:
Perihelion, where the planet is closest to the Sun, occurring at ﻿θ=0∘
.
Aphelion, where the planet is farthest from the Sun, occurring at ﻿θ=180∘
.
﻿
The orbit's shape is characterized by its eccentricity (﻿e
), a measure of how much the ellipse deviates from being circular. 
An eccentricity of 0 describes a perfect circle,
An eccentricity closer to 1 indicates a more elongated shape. 
This stretching means that the planet's distance from the Sun varies throughout its orbit.
Different planets have distinct eccentricities, influencing how elliptical their orbits are. See below for the ﻿e
 values for all the planets in our solar system.
﻿﻿Mercury
:0.2060​
﻿
﻿﻿Earth
:0.0167​
﻿
﻿﻿Uranus
:0.0472​
﻿
﻿﻿Jupiter
:0.0485​
﻿
﻿﻿Mars
:0.0934​
﻿
﻿﻿Pluto
:0.2500​
﻿
﻿﻿Venus
:0.0068​
﻿
﻿﻿Saturn
:0.0556​
﻿
﻿﻿Neptune
:0.0086​
﻿
ApplicationsThis calculator can be useful for:
Astronomers studying the motion of planets and other celestial bodies.
Space missions planning trajectories of spacecraft within the solar system.
Educational purposes to help students understand the dynamics of planetary orbits.
ReferencesMurray, C. D., & Dermott, S. F. (1999). Solar System Dynamics. Cambridge University Press.
Goldstein, H., Poole, C., & Safko, J. (2001). Classical Mechanics (3rd ed.). Addison-Wesley.
https://www.space.fm/astronomy/planetarysystems/keplers1stlaw.html
Chaisson, E., & McMillan, S. (2011). Astronomy: A Beginner's Guide to the Universe (7th ed.). Pearson.
https://www.britannica.com/science/Keplers-laws-of-planetary-motion
Related ResourcesKepler's Third Law Calculator
Y-coordinate of Keplerian Orbit
Orbital Mechanics Calculator
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