Definitions

# orbit

[awr-bit]
orbit, in astronomy, path in space described by a body revolving about a second body where the motion of the orbiting bodies is dominated by their mutual gravitational attraction. Within the solar system, planets, dwarf planets, asteroids, and comets orbit the sun and satellites orbit the planets and other bodies.

## Planetary Orbits

From earliest times, astronomers assumed that the orbits in which the planets moved were circular; yet the numerous catalogs of measurements compiled especially during the 16th cent. did not fit this theory. At the beginning of the 17th cent., Johannes Kepler stated three laws of planetary motion that explained the observed data: the orbit of each planet is an ellipse with the sun at one focus; the speed of a planet varies in such a way that an imaginary line drawn from the planet to the sun sweeps out equal areas in equal amounts of time; and the ratio of the squares of the periods of revolution of any two planets is equal to the ratio of the cubes of their average distances from the sun. The orbits of the solar planets, while elliptical, are almost circular; on the other hand, the orbits of many of the extrasolar planets discovered during the 1990s are highly elliptical.

After the laws of planetary motion were established, astronomers developed the means of determining the size, shape, and relative position in space of a planet's orbit. The size and shape of an orbit are specified by its semimajor axis and by its eccentricity. The semimajor axis is a length equal to half the greatest diameter of the orbit. The eccentricity is the distance of the sun from the center of the orbit divided by the length of the orbit's semimajor axis; this value is a measure of how elliptical the orbit is. The position of the orbit in space, relative to the earth, is determined by three factors: (1) the inclination, or tilt, of the plane of the planet's orbit to the plane of the earth's orbit (the ecliptic); (2) the longitude of the planet's ascending node (the point where the planet cuts the ecliptic moving from south to north); and (3) the longitude of the planet's perihelion point (point at which it is nearest the sun; see apsis).

These quantities, which determine the size, shape, and position of a planet's orbit, are known as the orbital elements. If only the sun influenced the planet in its orbit, then by knowing the orbital elements plus its position at some particular time, one could calculate its position at any later time. However, the gravitational attractions of bodies other than the sun cause perturbations in the planet's motions that can make the orbit shift, or precess, in space or can cause the planet to wobble slightly. Once these perturbations have been calculated one can closely determine its position for any future date over long periods of time. Modern methods for computing the orbit of a planet or other body have been refined from methods developed by Newton, Laplace, and Gauss, in which all the needed quantities are acquired from three separate observations of the planet's apparent position.

## Nonplanetary Orbits

The laws of planetary orbits also apply to the orbits of comets, natural satellites, artificial satellites, and space probes. The orbits of comets are very elongated; some are long ellipses, some are nearly parabolic (see parabola), and some may be hyperbolic. When the orbit of a newly discovered comet is calculated, it is first assumed to be a parabola and then corrected to its actual shape when more measured positions are obtained. Natural satellites that are close to their primaries tend to have nearly circular orbits in the same plane as that of the planet's equator, while more distant satellites may have quite eccentric orbits with large inclinations to the planet's equatorial plane. Because of the moon's proximity to the earth and its large relative mass, the earth-moon system is sometimes considered a double planet. It is the center of the earth-moon system, rather than the center of the earth itself, that describes an elliptical orbit around the sun in accordance with Kepler's laws. All of the planets and most of the satellites in the solar system move in the same direction in their orbits, counterclockwise as viewed from the north celestial pole; some satellites, probably captured asteroids, have retrograde motion, i.e., they revolve in a clockwise direction.

In physics, an orbit is the gravitationally curved path of one object around a point or another body, for example the gravitational orbit of a planet around a star.

Historically, the apparent motion of the planets were first understood in terms of epicycles, which are the sums of numerous circular motions. This predicted the path of the planets quite well, until Johannes Kepler was able to show that the motion of the planets were in fact elliptical motions. Sir Isaac Newton was able to prove that this was equivalent to an inverse square, instantaneously propagating force he called gravitation. Albert Einstein later was able to show that gravity is due to curvature of space-time, and that orbits lie upon geodesics and this is the current understanding.

## History

In the geocentric model of the solar system, mechanisms such as the deferent and epicycle were supposed to explain the motion of the planets in terms of perfect spheres or rings.

The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarized in his three laws of planetary motion. First, he found that the orbits of the planets in our solar system are elliptical, not circular (or epicyclic), as had previously been believed, and that the sun is not located at the center of the orbits, but rather at one focus. Second, he found that the orbital speed of each planet is not constant, as had previously been thought, but rather that the speed of the planet depends on the planet's distance from the sun. And third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the sun. For each planet, the cube of the planet's distance from the sun, measured in astronomical units (AU), is equal to the square of the planet's orbital period, measured in Earth years. Jupiter, for example, is approximately 5.2 AU from the sun and its orbital period is 11.86 Earth years. So 5.2 cubed equals 11.86 squared, as predicted.

Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies responding to an instantaneously propagating force of gravity were conic sections. Newton showed that a pair of bodies follow orbits of dimensions that are in inverse proportion to their masses about their common center of mass. Where one body is much more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.

Albert Einstein was able to show that gravity was due to curvature of space-time and was able to remove the assumption of Newton that changes propagate instantaneously. In relativity theory orbits follow geodesic trajectories which approximate very well to the Newtonian predictions. However there are differences and these can be used to determine which theory relativity agrees with. Essentially all experimental evidence agrees with relativity theory to within experimental measuremental accuracy.

## Planetary orbits

Within a planetary system; planets, dwarf planets, asteroids (a.k.a. minor planets), comets, and space debris orbit the central star in elliptical orbits. A comet in a parabolic or hyperbolic orbit about a central star is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. To date, no comet has been observed in our solar system with a distinctly hyperbolic orbit. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about that planet.

Owing to mutual gravitational perturbations, the eccentricities of the orbits of the planets in our solar system vary over time. Mercury, the smallest planet in the Solar System, has the most eccentric orbit. At the present epoch, Mars has the next largest eccentricity while the smallest eccentricities are those of the orbits of Venus and Neptune.

As two objects orbit each other, the periapsis is that point at which the two objects are closest to each other and the apoapsis is that point at which they are the farthest from each other. (More specific terms are used for specific bodies. For example, perigee and apogee are the lowest and highest parts of an Earth orbit, respectively.)

In the elliptical orbit, the center of mass of the orbiting-orbited system will sit at one focus of both orbits, with nothing present at the other focus. As a planet approaches periapsis, the planet will increase in speed, or velocity. As a planet approaches apoapsis, the planet will decrease in velocity.

### Understanding orbits

There are a few common ways of understanding orbits.

• As the object moves sideways, it falls toward the central body. However, it moves so quickly that the central body will curve away beneath it.
• A force, such as gravity, pulls the object into a curved path as it attempts to fly off in a straight line.
• As the object moves sideways (tangentially), it falls toward the central body. However, it has enough tangential velocity to miss the orbited object, and will continue falling indefinitely. This understanding is particularly useful for mathematical analysis, because the object's motion can be described as the sum of the three one-dimensional coordinates oscillating around a gravitational center.

As an illustration of an orbit around a planet, the Newton's cannonball model may prove useful (see image below). Imagine a cannon sitting on top of a tall mountain, which fires a cannonball horizontally. The mountain needs to be very tall, so that the cannon will be above the Earth's atmosphere and the effects of air friction on the cannonball can be ignored.

If the cannon fires its ball with a low initial velocity, the trajectory of the ball curves downward and hits the ground (A). As the firing velocity is increased, the cannonball hits the ground farther (B) away from the cannon, because while the ball is still falling towards the ground, the ground is increasingly curving away from it (see first point, above). All these motions are actually "orbits" in a technical sense — they are describing a portion of an elliptical path around the center of gravity — but the orbits are interrupted by striking the Earth.

If the cannonball is fired with sufficient velocity, the ground curves away from the ball at least as much as the ball falls — so the ball never strikes the ground. It is now in what could be called a non-interrupted, or circumnavigating, orbit. For any specific combination of height above the center of gravity, and mass of the planet, there is one specific firing velocity that produces a circular orbit, as shown in (C).

As the firing velocity is increased beyond this, a range of elliptic orbits are produced; one is shown in (D). If the initial firing is above the surface of the Earth as shown, there will also be elliptical orbits at slower velocities; these will come closest to the Earth at the point half an orbit beyond, and directly opposite, the firing point.

At a specific velocity called escape velocity, again dependent on the firing height and mass of the planet, an infinite orbit such as (E) is produced — a parabolic trajectory. At even faster velocities the object will follow a range of hyperbolic trajectories. In a practical sense, both of these trajectory types mean the object is "breaking free" of the planet's gravity, and "going off into space".

The velocity relationship of two objects with mass can thus be considered in four practical classes, with subtypes:

1. No orbit
2. Interrupted orbits
• Range of interrupted elliptical paths
3. Circumnavigating orbits
• Range of elliptical paths with closest point opposite firing point
• Circular path
• Range of elliptical paths with closest point at firing point
4. Infinite orbits
• Parabolic paths
• Hyperbolic paths

## Newton's laws of motion

In many situations relativistic effects can be neglected, and Newton's laws give a highly accurate description of the motion. Then the acceleration of each body is equal to the sum of the gravitational forces on it, divided by its mass, and the gravitational force between each pair of bodies is proportional to the product of their masses and decreases inversely with the square of the distance between them. To this Newtonian approximation, for a system of two point masses or spherical bodies, only influenced by their mutual gravitation (the two-body problem), the orbits can be exactly calculated. If the heavier body is much more massive than the smaller, as for a satellite or small moon orbiting a planet or for the Earth orbiting the Sun, it is accurate and convenient to describe the motion in a coordinate system that is centered on the heavier body, and we can say that the lighter body is in orbit around the heavier. (For the case where the masses of two bodies are comparable an exact Newtonian solution is still available, and qualitatively similar to the case of dissimilar masses, by centering the coordinate system on the center of mass of the two.)

Energy is associated with gravitational fields. A stationary body far from another can do external work if it is pulled towards it, and therefore has gravitational potential energy. Since work is required to separate two massive bodies against the pull of gravity, their gravitational potential energy increases as they are separated, and decreases as they approach one another. For point masses the gravitational energy decreases without limit as they approach zero separation, and it is convenient and conventional to take the potential energy as zero when they are an infinite distance apart, and then negative (since it decreases from zero) for smaller finite distances.

With two bodies, an orbit is a conic section. The orbit can be open (so the object never returns) or closed (returning), depending on the total kinetic + potential energy of the system. In the case of an open orbit, the speed at any position of the orbit is at least the escape velocity for that position, in the case of a closed orbit, always less. Since the kinetic energy is never negative, if the common convention is adopted of taking the potential energy as zero at infinite separation, the bound orbits have negative total energy, parabolic trajectories have zero total energy, and hyperbolic orbits have positive total energy.

An open orbit has the shape of a hyperbola (when the velocity is greater than the escape velocity), or a parabola (when the velocity is exactly the escape velocity). The bodies approach each other for a while, curve around each other around the time of their closest approach, and then separate again forever. This may be the case with some comets if they come from outside the solar system.

A closed orbit has the shape of an ellipse. In the special case that the orbiting body is always the same distance from the center, it is also the shape of a circle. Otherwise, the point where the orbiting body is closest to Earth is the perigee, called periapsis (less properly, "perifocus" or "pericentron") when the orbit is around a body other than Earth. The point where the satellite is farthest from Earth is called apogee, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the line-of-apsides. This is the major axis of the ellipse, the line through its longest part.

Orbiting bodies in closed orbits repeat their path after a constant period of time. This motion is described by the empirical laws of Kepler, which can be mathematically derived from Newton's laws. These can be formulated as follows:

1. The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of the ellipse. Therefore the orbit lies in a plane, called the orbital plane. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits around particular bodies; things orbiting the Sun have a perihelion and aphelion, things orbiting the Earth have a perigee and apogee, and things orbiting the Moon have a perilune and apolune (or, synonymously, periselene and aposelene). An orbit around any star, not just the Sun, has a periastron and an apastron.
2. As the planet moves around its orbit during a fixed amount of time, the line from Sun to planet sweeps a constant area of the orbital plane, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its perihelion than near its aphelion, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time."
3. For a given orbit, the ratio of the cube of its semi-major axis to the square of its period is constant.

Note that that while the bound orbits around a point mass, or a spherical body with an ideal Newtonian gravitational field, are all closed ellipses, which repeat the same path exactly and indefinitely, any non-spherical or non-Newtonian effects (as caused, for example, by the slight oblateness of the Earth, or by relativistic effects, changing the gravitational field's behavior with distance) will cause the orbit's shape to depart to a greater or lesser extent from the closed ellipses characteristic of Newtonian two body motion. The 2-body solutions were published by Newton in Principia in 1687. In 1912, Karl Fritiof Sundman developed a converging infinite series that solves the 3-body problem; however, it converges too slowly to be of much use. Except for special cases like the Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies.

Instead, orbits with many bodies can be approximated with arbitrarily high accuracy. These approximations take two forms.

One form takes the pure elliptic motion as a basis, and adds perturbation terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moon, planets and other bodies are known with great accuracy, and are used to generate tables for celestial navigation. Still there are secular phenomena that have to be dealt with by post-newtonian methods.

The differential equation form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces will equal the mass times its acceleration (F = ma). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial ones corresponds to solving an initial value problem. Numerical methods calculate the positions and velocities of the objects a tiny time in the future, then repeat this. However, tiny arithmetic errors from the limited accuracy of a computer's math accumulate, limiting the accuracy of this approach.

Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large objects have been simulated.

## Analysis of orbital motion

Please note that the following is a classical (Newtonian) analysis of orbital mechanics, which assumes the more subtle effects of general relativity (like frame dragging and gravitational time dilation) are negligible. General relativity does, however, need to be considered for some applications such as analysis of extremely massive heavenly bodies, precise prediction of a system's state after a long period of time, and in the case of interplanetary travel, where fuel economy, and thus precision, is paramount.

To analyze the motion of a body moving under the influence of a force which is always directed towards a fixed point, it is convenient to use polar coordinates with the origin coinciding with the center of force. In such coordinates the radial and transverse components of the acceleration are, respectively:

$a_r = frac\left\{d^2r\right\}\left\{dt^2\right\} - rleft\left(frac\left\{dtheta\right\}\left\{dt\right\} right\right)^2$
and
$a_\left\{theta\right\} = frac\left\{1\right\}\left\{r\right\}frac\left\{d\right\}\left\{dt\right\}left\left(r^2frac\left\{dtheta\right\}\left\{dt\right\} right\right)$.

Since the force is entirely radial, and since acceleration is proportional to force, it follows that the transverse acceleration is zero. As a result,

$frac\left\{d\right\}\left\{dt\right\}left\left(r^2frac\left\{dtheta\right\}\left\{dt\right\} right\right) = 0$.

After integrating, we have

$r^2frac\left\{dtheta\right\}\left\{dt\right\} = \left\{rm const.\right\}$

which is actually the theoretical proof of Kepler's 2nd law (A line joining a planet and the sun sweeps out equal areas during equal intervals of time). The constant of integration, h, is the angular momentum per unit mass. It then follows that

$frac\left\{dtheta\right\}\left\{dt\right\} = \left\{ h over r^2 \right\} = hu^2$
where we have introduced the auxiliary variable
$u = \left\{ 1 over r \right\}$.
The radial force is f(r) per unit mass is $a_r$, then the elimination of the time variable from the radial component of the equation of motion yields:

$frac\left\{d^2u\right\}\left\{dtheta^2\right\} + u = -frac\left\{f\left(1 / u\right)\right\}\left\{h^2u^2\right\}$.

In the case of gravity, Newton's law of universal gravitation states that the force is proportional to the inverse square of the distance:

$f\left(1/u\right) = a_r = \left\{ -GM over r^2 \right\} = -GM u^2$

where G is the constant of universal gravitation, m is the mass of the orbiting body (planet), and M is the mass of the central body (the Sun). Substituting into the prior equation, we have

$frac\left\{d^2u\right\}\left\{dtheta^2\right\} + u = frac\left\{ GM \right\}\left\{h^2\right\}$.

So for the gravitational force – or, more generally, for any inverse square force law – the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation (up to a shift of origin of the dependent variable). The solution is:

$u\left(theta\right) = frac\left\{ GM \right\}\left\{h^2\right\} + A cos\left(theta-theta_0\right)$
where $A$ and $theta_0$ are arbitrary constants.

The equation of the orbit described by the particle is thus:

$r = frac\left\{1\right\}\left\{u\right\} = frac\left\{ h^2 / GM \right\}\left\{1 + e cos \left(theta - theta_0\right)\right\}$,

where $e$ is:

$e equiv frac\left\{h^2A\right\}\left\{G M\right\} .$

In general, this can be recognized as the equation of a conic section in polar coordinates ($r$, $theta$). We can make a further connection with the classic description of conic section with:

$frac\left\{h^2\right\}\left\{GM\right\} = a\left(1-e^2\right)$

If parameter $e$ is smaller than one, $e$ is the eccentricity and $a$ the semi-major axis of an ellipse.

## Orbital planes

The analysis so far has been two dimensional; it turns out that an unperturbed orbit is two dimensional in a plane fixed in space, and thus the extension to three dimensions requires simply rotating the two dimensional plane into the required angle relative to the poles of the planetary body involved.

The rotation to do this in three dimensions requires three numbers to uniquely determine; traditionally these are expressed as three angles.

## Orbital period

The orbital period is simply how long an orbiting body takes to complete one orbit.

## Specifying orbits

It turns out that it takes a minimum 6 numbers to specify an orbit about a body, and this can be done in several ways. For example, specifying the 3 numbers specifying location and 3 specifying the velocity of a body gives a unique orbit that can be calculated forwards (or backwards). However, traditionally the parameters used are slightly different.

The traditionally used set of orbital elements is called the set of Keplerian elements, after Johannes Kepler and his Kepler's laws. The Keplerian elements are six:

In principle once the orbital elements are known for a body, its position can be calculated forward and backwards indefinitely in time. However, in practice, orbits are affected or perturbed, by forces other than gravity due to the central body and thus the orbital elements change over time.

## Orbital perturbations

An orbital perturbation is when a force or impulse which is much smaller than the overall force or average impulse of the main gravitating body and which is external to the two orbiting bodies causes an acceleration, which changes the parameters of the orbit over time.

It can be shown that a radial impulse given to a body in orbit doesn't change the orbital period (since it doesn't affect the angular momentum), but changes the eccentricity. This means that the orbit still intersects the original orbit in two places.

For a prograde or retrograde impulse (i.e. an impulse applied along the orbital motion), this changes both the eccentricity as well as the orbital period, but any closed orbit will still intersect the perturbation point. Notably, a prograde impulse given at periapsis raises the altitude at apoapsis, and vice versa, and a retrograde impulse does the opposite.

A transverse force out of the orbital plane causes rotation of the orbital plane.

### Orbital decay

If some part of a body's orbit enters an atmosphere, its orbit can decay because of drag. Particularly at each periapsis, the object scrapes the air, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum. This is similar to the effect of slowing a pendulum at its lowest point; the highest point of the pendulum's swing becomes lower. With each successive slowing more of the orbit's path is affected by the atmosphere and the effect becomes more pronounced. Eventually, the effect becomes so great that the maximum kinetic energy is not enough to return the orbit above the limits of the atmospheric drag effect. When this happens the body will rapidly spiral down and intersect the central body.

The bounds of an atmosphere vary wildly. During solar maxima, the Earth's atmosphere causes drag up to a hundred kilometres higher than during solar minima.

Some satellites with long conductive tethers can also decay because of electromagnetic drag from the Earth's magnetic field. Basically, the wire cuts the magnetic field, and acts as a generator. The wire moves electrons from the near vacuum on one end to the near-vacuum on the other end. The orbital energy is converted to heat in the wire.

Orbits can be artificially influenced through the use of rocket motors which change the kinetic energy of the body at some point in its path. This is the conversion of chemical or electrical energy to kinetic energy. In this way changes in the orbit shape or orientation can be facilitated.

Another method of artificially influencing an orbit is through the use of solar sails or magnetic sails. These forms of propulsion require no propellant or energy input other than that of the sun, and so can be used indefinitely. See statite for one such proposed use.

Orbital decay can also occur due to tidal forces for objects below the synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises tidal bulges in the primary, and since below the synchronous orbit the orbiting object is moving faster than the body's surface the bulges lag a short angle behind it. The gravity of the bulges is slightly off of the primary-satellite axis and thus has a component along the satellite's motion. The near bulge slows the object more than the far bulge speeds it up, and as a result the orbit decays. Conversely, the gravity of the satellite on the bulges applies torque on the primary and speeds up its rotation. Artificial satellites are too small to have an appreciable tidal effect on the planets they orbit, but several moons in the solar system are undergoing orbital decay by this mechanism. Mars' innermost moon Phobos is a prime example, and is expected to either impact Mars' surface or break up into a ring within 50 million years.

Finally, orbits can decay via the emission of gravitational waves. This mechanism is extremely weak for most stellar objects, only becoming significant in cases where there is a combination of extreme mass and extreme acceleration, such as with black holes or neutron stars that are orbiting each other closely.

### Oblateness

The standard analysis of orbiting bodies assumes that all bodies consist of uniform spheres, or more generally, concentric shells each of uniform density. It can be shown that such bodies are gravitationally equivalent to point sources.

However, in the real world, many bodies rotate, and this introduces oblateness and distorts the gravity field, and gives a quadrupole moment to the gravitational field which is significant at distances comparable to the radius of the body.

The general effect of this is to change the orbital parameters over time; predominantly this gives a rotation of the orbital plane around the rotational pole of the central body (it perturbs the argument of perigee) in a way that is dependent on the angle of orbital plane to the equator as well as altitude at perigee.

### Other gravitating bodies

The effects of other gravitating bodies can be very large. For example, the orbit of the Moon cannot be in any way accurately described without allowing for the action of the Sun's gravity as well as the Earth's.

### Light radiation and stellar wind

For small bodies particularly, light and stellar wind can cause significant perturbations to the attitude and direction of motion of the body, and over time can be quite significant.

## Scaling in gravity

The gravitational constant G is measured to be:

• (6.6742 ± 0.001) × 10−11 N·m²/kg²
• (6.6742 ± 0.001) × 10−11 m³/(kg·s²)
• (6.6742 ± 0.001) × 10−11 (kg/m³)-1s-2.

Thus the constant has dimension density-1 time-2. This corresponds to the following properties.

Scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence orbital periods remain the same. Similarly, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the earth.

When all densities are multiplied by four, orbits are the same, but with orbital velocities doubled.

When all densities are multiplied by four, and all sizes are halved, orbits are similar, with the same orbital velocities.

These properties are illustrated in the formula (known as Kepler's 3rd Law)

$GT^2 sigma = 3pi left\left(frac\left\{a\right\}\left\{r\right\} right\right)^3,$

for an elliptical orbit with semi-major axis a, of a small body around a spherical body with radius r and average density σ, where T is the orbital period.