How To Calculate Perihelion

In astrophysics, the ​perihelion​ is the point in an object's orbit when it is closest to the sun. It comes from the Greek for near (​peri​) and sun (​Helios​). Its opposite is the ​aphelion​, the point in its orbit at which an object is furthest from the sun.

The concept of perihelion is probably most familiar in relation to ​comets​. The orbits of comets tend to be long ellipses with the sun situated at one focal point. As a result, most of the comet's time is spent far away from the sun.

However, as comets approach perihelion, they get close enough to the sun that its heat and radiation cause the approaching comet to sprout the bright coma and long glowing tails that make them some of the most famous celestial objects.

Read on to learn more about how perihelion relates to orbital physics, including a ​perihelion​ formula.

Although many of us carry an idealized image of the Earth's path around the sun as a perfect circle, the reality is very few (if any) orbits are actually circular – and Earth is no exception. Almost all of them are actually ​ellipses​.

Astrophysicists describe the difference between an object's hypothetically perfect, circular orbit and its imperfect, elliptical orbit as its ​eccentricity​. Eccentricity is expressed as a value between 0 and 1, sometimes converted to a percentage.

An eccentricity of zero indicates a perfectly circular orbit, with larger values indicating increasingly elliptical orbits. For instance, the Earth's not-quite-circular orbit has an eccentricity of about 0.0167, whereas the extremely elliptical orbit of Halley's comet has an eccentricity of 0.967.

When talking about orbital motion, it's important to understand some of the terms used to describe ellipses:

•​foci​: two points inside the ellipse that characterize its shape. Foci that are closer together mean a more circular shape, farther apart mean a more oblong shape. When describing solar orbits, one of the foci will always be the sun.
•​center​: every ellipse has one center point.
•​major axis​: a straight line across the longest width of the ellipse, it passes through both foci and the center, its endpoints are the vertices.
•​semi-major axis​: half of the major axis, or the distance between the center and one vertices.
•​vertices​: the point at which an ellipse makes its sharpest turns and the two furthest points from each other in the ellipse. When describing solar orbits, these correspond to the perihelion and aphelion.
•​minor axis​: a straight line cross the shortest width of the ellipse, it passes through the center. It endpoints are the co-vertices.
•​semi-minor axis:​ half of the minor axis, or the shortest distance between the center and a co-vertex of the ellipse.

If you know the length of an ellipse's major and minor axes, you can calculate its eccentricity using the following formula:

\(\text{eccentricity}^2 = 1.0-\frac{\text{semi-minor axis}^2}{\text{semi-major axis}^2}\)

Typically, lengths in orbital movement are measured in terms of astronomical units (AU). One AU is equal to the mean distance from the center of the Earth to the center of the sun, or ​149.6 million kilometers​. The specific units used to measure the axes don't matter as long as they are the same.

With all that out of the way, calculating perihelion and aphelion distances is actually quite easy as long as you know the length of an orbit's ​major axis​ and its ​eccentricity​. Use the following formula:

\(\text{perihelion} = \text{semi-major axis}(1-\text{eccentricity})\text{ }\)
\(\text{aphelion} =\text{semi-major axis}(1 + \text{eccentricity})\)

Mars has a semi-major axis of 1.524 AU and a low eccentricity of 0.0934, therefore:

\(\text{perihelion}_{Mars} = 1.524\text{ AU}(1-0.0934)=1.382\text{ AU}\text{ }\)
\(\text{aphelion}_{Mars} =1.524\text{ AU}(1 + 0.0934)=1.666\text{ AU}\)

Even at the most extreme points in its orbit, Mars remains roughly the same distance from the sun.

Earth, likewise, has a very low eccentricity. This helps keep the planet's supply of solar radiation relatively consistent throughout the year and means that Earth's eccentricity doesn't have an extremely noticeable impact on our day-to-day lives. (The tilt of the earth on its axis has a much more noticeable effect on our lives by causing the existence of seasons.)

Now let's calculate the perihelion and aphelion distances of Mercury from the sun instead. Mercury is much closer to the sun, with a semi-major axis of 0.387 AU. Its orbit is also considerably more eccentric, with an eccentricity of 0.205. If we plug these values into our formulas:

\(\text{perihelion}_{Mercury} = 0.387\text{ AU}(1-0.206)=0.307\text{ AU}\text{ }\)
\(\text{aphelion}_{Mercury} =0.387\text{ AU}(1 + 0.206)=0.467\text{ AU}\)

Those numbers mean that Mercury is almost ​two thirds​ closer to the sun during perihelion than it is at aphelion, creating much more dramatic changes in how much heat and solar radiation the sunward surface of the planet is exposed to over the course of its orbit.