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{ Practical astronomy | Astronomy | Planets | Planetary motion }

Planetary motion

Ephemerides are ephemeral data describing the motion of a celestial body. Simply put, they are a table of times, and the celestial position and distance at that time. Ephemeris and ephemerides are the singular and plural of the word, but ephemeris are also not really countable, so that both terms are used interchangeably and essentially have the same meaning.

As Earthlings we need ephemeris to tell us where in our sky the planets will be at a given time. But the movement in question is actually a combination of the planet's movement around the Sun and of the Earth around the Sun.

The elliptical Kepler orbit
The elliptical Kepler orbit. For demonstration purposes, the eccentricity here is rather extreme at e = 0.6.

What is going on is discussed in detail in the section about the Sun. The Sun's gravity keeps each planet in an orbit around it. If the planet and Sun were alone with each other, and in Newtonian mechanics, this would be a Kepler orbit, i.e. an ellipse. However, each pair of planets also attracts each other a tiny bit, gravity travels between Sun and planet only at the speed of light, and Einstein has his own ideas of space and time. As a result, the Kepler ellipse is actually not precise enough for good ephemeris of the planets.

In the section about the Sun we presented the algorithm by Simon et al. (1994a) to take care of these problems. Our deliberations there were for the Earth's orbit around the Sun, although applied as the apparent orbit of the Sun around the Earth. The whole formalism applies for each of the other seven planets and its motion around the Sun, just with different parameters for their orbits.

The calculations still use Kepler elements and the iteration of the Kepler equation; it is just that the Kepler elements are not constant, but adjusted beforehand for the time in question. We recall here that the Kepler elements are:

Semi-major axis (in Gm or au). This is the long radius of the ellipse, also the average distance of the planet from the Sun.
Numeric eccentricity (dimensionless). This is zero for a circle, one for a parabola, and in between characterises how non-circular the ellipse is.
Mean daily motion (in °/d).
Argument of perihelion (in °). This is the angle between the ascending node and the perihelion and describes the orientation of the ellipse in the orbital plane.
Inclination (in °). This is the inclination angle between the ecliptic and the orbital plane.
Longitude of ascending node (in °). This is the angle between the vernal equinox and the ascending node (where the orbital plane intersects the ecliptic).
An inclined Kepler orbit
An inclined Kepler orbit (Meyerdierks 1982a).

In addition, time is parameterised as the mean anomaly M. Its value at a certain standard epoch T0 is usually listed as an orbital element and increases according to the mean daily motion n.

M(t) = M(T0) + n · (t − T0)

We briefly mention again, that in the literature we usually find ϖ = ω + Ω and L = M +ω + Ω. Those are less intuitive and also less useful for the calculations.

Because we want to know where the planet appears in the sky above Earth, we have to make calculations for two planets, the one being observed (P) and the Earth (E).

ME(t) = ME(T0,E) + nE · (t − T0,E)
MP(t) = MP(T0,P) + nP · (t − T0,P)

The Kepler equation

EX − eX · sin(EX) = MX

is solved iteratively to find the eccentric anomalies EE(t) and EP(t). From the angle E (see graph of the Kepler orbit for its meaning), we can calculate the rectangular, heliocentric coordinates in the orbital plane, oriented to the perihelion:

xS,X = aX · cos(EX) − aX · eX
yS,X = aX · sin(EX) · (1 − eX2)0.5
zS,X = 0

coordinate translation (origin shift)
Mars in heliocentric (x1,y1,z1) and geocentric coordinates (x2,y2,z2).

The remaining Kepler elements ω, i and Ω are just rotations of the coordinate system in space.

x1 = xS,X · cos(ωX) − yS,X · sin(ωX)
y1 = yS,X · cos(ωX) + xS,X · sin(ωX)
z1 = zS,X

x2 = x1
y2 = y1 · cos(iX) − z1 · sin(iX)
z2 = z1 · cos(iX) + y1 · sin(iX)

xhel,X = x2 · cos(ΩX) − y2 · sin(ΩX)
yhel,X = y2 · cos(ΩX) + x2 · sin(ΩX)
zhel,X = z2

Having made the above calculations separately for Earth and planet, we finally find the geocentric position of the planet as the difference of the two heliocentric positions:

xgeo,P = xhel,P − xhel,E
ygeo,P = yhel,P − yhel,E
zgeo,P = zhel,P − zhel,E

Inner planets

A synodic orbit of Mercury
A synodic orbit of Mercury from one superior conjunction (right) to the next (left).

The motion of a planet in the Earth's sky appears quite complex, because of the combination of the planet's motion around the Sun and of the Earth's motion around the Sun. The inner planets Mercury and Venus remain close to the Sun in our sky; Mercury is at most 28° from the Sun, Venus 47°. As such, observation of an inner planet is mostly limited to twilight or daylight. A synodic orbit of an inferior planet has four main events: superior and inferior conjunctions and greatest eastern and western elongations.

Outer planets

A synodic orbit of Jupiter
A synodic orbit of Jupiter from one conjunction (right) to the next (left).

The apparent motion of an outer planet on our sky is very different. Such a planet is not limited to the vicinity of the Sun, but can appear in the night sky and even directly opposite the Sun. The general movement of such a planet in our sky is direct or eastward. However, surrounding opposition, the faster parallel motion of the Earth reflects against the background of the stars as a retrograde motion. You can observe something similar when overtaking a slower car on the road and observing its appearance in front of the landscape beyond. The synodic orbit of an outer planet has four main events: conjunction, opposition and two standstill times either side of opposition.

For all planets, the distance from Earth changes by about twice the Earth's distance from the Sun. This is a more significant change for our neighbours Venus and Mars, also for Mercury. But Jupiter and beyond, are at least 5 times farther from the Sun than Earth is, and the change of distance during their synodic orbit is small. Connected with this, their synodic orbits last not much longer than one year. For our neighbours, Venus and Mars the synodic orbit lasts much longer, just under two years for Venus and just over two years for Mars.