Charting the Heavens

By Mohit Mahapatra, BS 2022. 

Any aspiring wayfarer of the stars must know how to chart the celestials. Whether you have grand old single-use rockets that are 90% fuel and propellant (like we had in the 20th century and still use), ultra-efficient torchships (à la the Expanse) or even engines faster than light (causality, what’s that?), you cannot do without the basics of astrography - orbital elements.

Put simply, orbital elements are a set of parameters that completely define a celestial object’s “coordinates” or its orbit around another body. Gravity dominates at astronomical scales, and every object in a stable orbit moves through an elliptical path. Using these elements, a spacefarer can not only pinpoint the body’s location without seeing it but even predict its position at a future point in time or determine where it would’ve been some time in the past. Quite nifty, huh?

Humanity has been fascinated by the vast black void since before they knew it was a vast black void. Though the ancients (and not-so-ancients) could not leave Earth’s gravity well, they could observe the skies and make models of celestial motion even before they knew what gravity and orbits were.

Celestial spheres - outdated but effective (for the time) models of celestial motion

The celestial spheres of the past (not to be confused with the celestial sphere, which is an abstract concept still in use and similar in fashion) were developed to describe the motion of objects using the only reference point at the time - Earth. While these would still be useful to some extent while mapping the sky of a single planet or moon, the march of progress has given us far better tools that are not only simpler and more accurate but also more useful for a star-hopping spacefarer.

Enter modern orbital elements. With the knowledge of gravity and the laws of motion in the heavens, we have developed the field of celestial mechanics which allows us to describe and predict the motion of objects to a great level of accuracy. Though not perfect as they consider a two-body system and can give a slightly different result due to the effects of the gravity of other bodies and also due to relativity, the classical elements are an essential tool for any traveler in the void.

Johannes Kepler and Tycho Brahe - two astronomers very relevant for a starfarer

First of all, a starfarer must understand orbits. We can’t describe their characteristics without knowing what they are, can we? But fear not, German astronomer Johannes Kepler has you covered.

Born in 1571 in the Holy Roman Empire, Johannes Kepler was also an astrologer, natural philosopher (studying an archaic form of physics that included biology and chemistry as well), mathematician and writer. Working on Tycho Brahe’s planetary observation data for 5 years from 1601, Kepler published the laws he’s famed for in 1609. His three laws of planetary motion are elementary for a starfarer’s success.

1.   A planet’s orbit around a star is an ellipse.

2.  The star is at one of the foci of the ellipse.

3.  The area (of the elliptic sector) swept by the planet is fixed for a given time interval throughout its orbit.

With these laws of motion, we can approximate celestial bodies as following Kepler orbits. These can be ellipses, parabolas, hyperbolas or theoretically even straight lines (the last one not seen in practice as the two bodies would collide, which never goes well). And now we can proceed to defining all aspects of these orbits.

An orbit has four important parts a starfarer should be acquainted with lest they be cursed with having to check a dictionary every minute:

• Primary: A usually quite massive object being orbited by another object.

• Secondary: An object orbiting another, more massive celestial body.

• Periapsis: The point in a secondary’s orbit closest to the primary.

• Apoapsis: The point in a secondary’s orbit farthest from the primary

The center of mass of the primary is approximated as the center of mass of the two-body system. However, even if the primary and secondary have comparable masses, they can be described using two separate Kepler orbits. In this case, the center of mass (the “primary”) becomes the barycenter of the two-body system.

Named after Johannes Kepler and the laws of planetary motion, Keplerian elements are the traditional set of 6 parameters used to uniquely define the orbit of a celestial body around another. Old and reliable, these parameters are delightfully useful for a spacefarer. Not only can they describe the motion of planets, moons and asteroids, but also that of your starfaring spacecraft and artificial satellites!

As we proceed, Pluto will be our example. Formerly the ninth planet of Sol and demoted to dwarf planet status by a redefinition of the word ‘planet’ by the IAU (International Astronomical Union) in 2006, Pluto nevertheless remains an interesting world hiding many secrets under its icy layers. And knowing about its orbit can be a great learning experience for any spacefarer.

The first parameter concerning any starfarer is the size of orbit. This is given by the semi-major axis (SMA), denoted by ‘a’. It is simply half the length of the longest line that can fit in the orbit. The SMA determines many important things such as how much thrust you might require to enter the body’s orbit or how long its year would last, if it is a large object like a planet.

a = periapsis distance + apoapsis distance

First Keplerian element: semi-major axis (a)

Pluto: a = 5.906 billion km or 39.48 AU

With an SMA over 39 times greater than that of Earth, Pluto is among the most distant places in the solar system from our homeworld. This gives Pluto a year of about 250 Earth days, which is over 366 Pluto days. Huh, that sounds familiar…

Now we know the size of the orbit. But what use is the size without the shape? What if you’re on one of those dreaded straight-line orbits? You won’t even notice until you plunge into the star as you sleep!

The second parameter is the eccentricity ‘e’ of the orbit. This measures how elongated or flattened the orbit is compared to a circle. The infamous straight line has an eccentricity of 1, such that the primary (which could be a star) is situated at one end of the line segment and the secondary (which could be your expensive spaceship) goes through that point. It is recommended to rely on robust orbital mechanics know-how instead of spacecraft insurance.

        e = distance b/w center & focus ÷ semi-major axis                  

Components of second Keplerian element: eccentricity (e)

Pluto: e = 0.2488

A wildly eccentric orbit means that at its closest, Pluto is 29.66 AU away from the sun (which is closer than Neptune’s closest approach!) while at its furthest the icy dwarf is over 49.3 AU away from the sun.

Space is a lonely place, and a friendly face is hard to trace. If you want to meet up with a fellow wanderer of the stars, the Keplerian elements are the perfect tool! But what if your friend’s orbit is tilted compared to yours? It could be years before you two can meet! The third element comes to the rescue.

The third parameter is the tilt of the object with respect to a reference plane, which is usually the equatorial plane of the primary object. This is the inclination ‘i’. It can actually be quite hard (in terms of propulsion) to meet up with a friend having the exact same orbit but a different tilt.

i = angle b/w reference plane & orbital plane

(measured anticlockwise on the plane perpendicular to the two planes)

Third Keplerian element: inclination (i)

Pluto: i = 17.16° to the ecliptic or 11.88° to the sun’s equator

With a very high orbital inclination compared to the major planes, Pluto dips in and out of the ecliptic twice each slow Pluto year. This can make getting to Pluto slightly tricky for a new spacefarer, requiring an orbital inclination change maneuver. The ecliptic is roughly the plane of Earth’s orbit.

At this point in the guide with newly-gained information about orbital elements, new starfarers often schedule meetings in Sol with Pluto’s orbit as the venue. What other detail could they need? And then they reach their meeting point, and almost a day later receive the message:

“What are you doing on the other side of the sun? Come over here!”

This is a natural result of not considering the fourth parameter. That is the swivel of the orbit, or its horizontal rotation on the reference plane keeping the tilt fixed. This is the longitude (or right ascension) of the ascending node ‘Ω’.  Without this information, the part of Pluto’s orbit that sticks out of the ecliptic could be anywhere, and calling the guide terrible before reading all of it isn’t a fix.

Ω = angle b/w ascending node vector & reference direction

(measured anticlockwise in the reference plane)

Fourth Keplerian element: longitude of ascending node (Ω)

Pluto: Ω = 110.299°

 If Earth had an orbit that was as tilted as that of Pluto, there would be a longitude difference of over 120°, which would require its own set of spacecraft maneuvers. New starfarers need not fear though, as the Tome has you covered! For just 499 Stellar Credits you can buy the newest edition of our - *ahem*, moving on -

Orbital racing is a favorite pastime of readers of the Starfarer’s Tome. Contrary to the initial thought it evokes, it is a game of finesse and not engine power. Competitors follow a celestial object’s orbit closely around the star and try to reach its apoapsis or periapsis from a starting point.

Those using constant thrusts to adjust their velocity instead of orbital mechanics knowledge often stray too far and lose. Needless to say, do not try this game with Pluto unless you’re interested in posthumous awards.

The fifth parameter is the argument of periapsis ‘ω’ helps to a great extent in this elaborate dance of a race. It defines the orientation of the orbit in the orbital plane.

  ω = angle b/w ascending node vector & periapsis

(measured anticlockwise in the orbital plane)

Fifth Keplerian element: argument of periapsis (ω)

Pluto: ω = 113.834°

There is a less than 0.5° difference between Pluto and Earth’s periapsis vectors. This means that you could draw a pretty straight-looking line through their closest points in orbit to the sun. No, this has no astrological significance.

With the five elements covered thus far, the orbit itself has been defined completely. However, the secondary itself, at any given point of time, could be at any point in the orbit. For a starfarer, two objects in the exact same orbit but at different points in that orbit essentially have different orbits for all practical purposes, as there is another set of maneuvers to go from one point to another in the same orbit.

The last parameter gives the position of the object at epoch, with the currently used standard being J2000 (1st January 12:00 Terrestrial Time by the Gregorian calendar). This is the true anomaly ‘ν’.

ν = angle b/w periapsis & position vector at epoch

(measured anticlockwise in the orbital plane)

A similar quantity is mean anomaly (‘M’). It gives the angle a similar secondary object would have moved if it had a circular orbit instead of an elliptical one. It is a convenient angle without a real geometric significance that varies with time.

Sixth Keplerian element: true anomaly (ν)

Pluto: M = 14.53°

Not much to say here, other than the fact that if all the other orbital elements were equal, Pluto would very roughly be on the other side of the sun from Earth. Coincidence? I think n - yes.

Putting the six elements together, the starfarer’s collection of coordinates is finally complete. You can now go on a celestial voyage without getting lost or stranded!

The six elements

Sources:

• https://science.nasa.gov/resource/orbits-and-keplers-laws/ 

• https://www.classe.cornell.edu/~seb/celestia/orbital-parameters.html 

• https://www.faa.gov/sites/faa.gov/files/about/office_org/headquarters_offices/avs/III.4.1.4_Describing_Orbits.pdf 

• Astronomia Nova, Johannes Kepler

• Harmonices Mundi, Johannes Kepler

• Epitome Astronomiae Copernicanae, Johannes Kepler