# Can we determine the position of an artificial satellite using Kepler's parameters?

I have read that an orbit is defined by Kepler's parameters, so is it possible to determine an exact position of an artificial satellite using those parameters?

Forgive me if this is a redundant question, I am new to this field, I came across this and was really fascinated by this. Thankyou for your time.

• Re exact position - Not at all. Exactness is an impossibility. There are always errors. Always. Even for an artificial satellite in Earth orbit that is outfitted with GPS, there are errors. Keplerian orbits are a nice approximation, an approximation that gets ever worse as time goes on. Mar 16, 2021 at 17:20

There are 6 keplerian orbital elements (but note that these parameters are not the only way to describe an orbit). Semi-major axis, eccentricity, inclination, argument of periapsis, right ascension, and true anomaly. Here is a surface level explanation of each:

Semi-major axis describes the "size" of the orbit. In the case of circular orbits (or orbits with small eccentricities), the semi-major axis describes how large the orbit is (distance from the central body, in a lot of cases its Earth).

Eccentricity describes the shape of the orbit. e = 0 means the orbit is circular. e = 1 means the orbit is parabolic (it has just enough energy to reach an infinite distance away from Earth), and e > 1 means its hyperbolic (has excess velocity on top of escape velocity).

Inclination, argument of periapsis, and right ascension together describe the orientation of the orbital plane with respect to the Earth equatorial inertial frame (if we are assuming Earth orbits). They do so by a 3-1-3 euler angle sequence, where each angle corresponds to one of those rotations. I can elaborate more if you'd like, but that seems out of the scope of your question.

And finally the true anomaly describes the angle between the orbit periapsis and the current position of the orbiting body, which is what you're looking for.

In case you're wondering why its called "true" anomaly, its because there also exists eccentric and mean anomalies, which have different geometrical representations of where an object is in its orbit.

This wikipedia article has a good visual on the elements: https://en.wikipedia.org/wiki/Orbital_elements

• Well explained, is there any algorithm/theory/formula which quantifies all the parameters to get the exact location of the satellite? Mar 13, 2021 at 20:03
• I'm a bit unsure what you mean, but the answer is yes. If by location you mean 3D vectors of position, you can always convert between keplerian orbital elements and 3D position and velocity vectors (and vice versa). You can also from there calculate the position of any object with respect to the surface of the Earth (if you want to observe an object with a telescope or antenna) Mar 13, 2021 at 20:24
• Yes, thank you for the correction! I just updated my comment Mar 13, 2021 at 20:59
• This answer is flat out wrong. Mar 16, 2021 at 17:13

Can we determine the position of an artificial satellite using Kepler's parameters?

The Earth's gravitational field is not uniform, and the Earth's atmosphere, the Sun, the Moon, and the other planets perturb the orbits of artificial satellites that orbit the Earth.

Multiple satellites explicitly take advantage of the Earth's non-uniform gravitational field. Examples of these include the sun-synchronous satellites, of which there are many. A Keplerian orbital plane does not change. The only orbital element that changes in Keplerian dynamics is the orbiting object's true anomaly. The oblateness of the Earth causes the orbital planes of Earth orbiting satellites to precess. Designers of sun-synchronous satellites choose just the right altitude and just the right inclination so as to make the perturbations caused by the Earth's oblateness make the orbital planes of those satellites precess in manner that keeps the satellite in sync with sunrise / sunset.

Another example of non-Keplerian artificial satellite orbits is the very important concept of geosynchronous satellites. The orbits of those satellites would be very simple to maintain if they followed Keplerian orbits. But they don't. Geosynchronous satellites necessarily carry propellant so as to occasionally readjust their orbits to counteract perturbations from the Earth's non-spherical shape and from third body influences such as the Moon and the Sun.

Even the orbits of the planets are not quite Keplerian. Each of the Keplerian elements that should be constant are not quite constant regarding the Earth's orbit about the Sun. The variations are part of why the Earth occasionally undergoes ice ages.

The Earth's Moon has a markedly non-Keplerian orbit. Understanding the orbit of the Moon was a very long term endeavor, and it wasn't explained nicely until the tail end of the 19th century, two centuries after Newton first published his Principia (and even more time after Kepler published his works). And that explanation did not take into account the recession of the Moon from the Earth.

Mercury also has a markedly non-Keplerian orbit. It's argument of periapsis precesses by over 500 arcseconds per century, mostly due to the influences of the other planets. Even after accounting for those perturbations, there's a 43 arcsecond per century precession that Newtonian mechanics cannot explain. This 43 arcsecond per century precession discrepancy resulted in a failed hunt for a planet (Vulcan) that did not exist and was only explained by general relativity. The extremely close match between the observed discrepancy and the theoretical post-diction by general relativity was one of the key reasons general relativity was accepted so quickly.

• Every object in space has a non-keplerian orbit thats true, since the elements describe an orbit at one point in time using 2 body dynamics. But that isn't the question. Yes the keplerian orbital elements can tell you exactly where an object is at a given time. How long they are valid for is a different question. One common way to get the "location" of your satellite is through TLEs, which give you keplerian orbital elements. You can propagate those over time using whatever perturbations you want, but you still describe the orbit with the elements Mar 16, 2021 at 14:09
• @AlfonsoGonzalez TLEs are not Keplerian elements. They specifically account for (or to account for) the Earth's oblateness, drag due to the Earth's atmosphere, and solar radiation pressure. Mar 16, 2021 at 14:29
• TLEs give you inclination, RAAN, eccentricity, argument of perigee, mean anomaly and mean motion, which you can then calculate semi-major axis and true anomaly from. So yes TLEs give you keplerian elements. They also give first and second derivative of mean motion and B* drag term, which can then be used to propagate the orbit with perturbations. SGP4 is a popular propagator which takes in the TLE and uses a simplified perturbation model to propagate given that info. So I believe my response still stands. Mar 16, 2021 at 14:40
• Link to space-track.org documentation on TLE format: space-track.org/documentation#tle Mar 16, 2021 at 14:40
• @AlfonsoGonzalez Two downvotes for a correct answer, versus three upvotes for your incorrect answer? What is going on here? Just because TLEs have the same name as those used for Keplerian elements does not mean that TLEs are Keplerian elements. They are not. They are instead specifically designed for use with the SGP4 algorithm, which explicitly accounts for the non-Keplerian nature of satellites that orbit the Earth. Mar 16, 2021 at 17:13