Orbit analysis

The orbit analysis tool provides information about the current trajectory, seen as a perturbed Kepler orbit.

Overview

The orbit analysis tool analyses the current trajectory of the active vessel, as an orbit around an automatically detected primary. The trajectory is analysed over a given mission duration.

The computation occurs in the background; it is tracked by a progress bar.

Below the progress bar is the orbit analysis report from the last computation.

Contents

• Lowest altitude
• Orbital elements
  • Graphs
  • Orbit stability
    • Frozen orbits about an oblate body
    • The von Zeipel–Лидов–古在 mechanism
• Recurrence properties

Lowest altitude

The tool indicates the lowest altitude that will be reached over the mission duration, and indicates whether the spacecraft will reenter, risk collision (that is, reach an altitude lower than that of the highest terrain), or certainly collide within the mission duration.

Example on an orbit that comes dangerously low:

Orbital elements

Periods

Three periods are given.

• The sidereal period is the period of the mean longitude $Ω+ω+M$.
• The nodal period is the period of the (mean*) mean argument of latitude $ω+M$. This approximately corresponds to the time between successive ascending nodes.
• The anomalistic period is the period of the (mean*) mean anomaly $M$. This approximately corresponds to the time between successive periapsides.

In a Kepler orbit, the longitude of the ascending node $Ω$ and the argument of the periapsis ω are constant, so that all these periods are the same; this is not the case when the orbit is perturbed, and both $Ω$ and $ω$ vary.

* These elements are a moving average over one sidereal period of the orbital elements called the mean argument of latitude $ω+M$ and the mean anomaly $M$, so they are mean versions of these elements.

Example on a Молния orbit with RSS:

Mean elements

The elements given here are free from short-period variations, that is, variations whose period is approximately one revolution. This is in contrast to the osculating elements (those shown, e.g., by the MechJeb Orbit Info window, or the Kerbal Engineer Redux orbital display), which will exhibit strong periodic variations over a revolution.

While short-period variations are eliminated, long-period and secular variations remain in these elements. In order to express the magnitude of these variations, the elements are given as a range expressed as midpoint±half-width, indicating that the mean element remains within half-width of midpoint over the mission duration.

The following elements are displayed:

• the mean semimajor axis $a$;
• the mean eccentricity $e$;
• the mean inclination $i$;
• the mean longitude of the ascending node $Ω$;
• the mean argument of the periapsis $ω$.

The precession rate of $Ω$ (the rate of nodal precession, i.e., how fast ascending node moves along the equator) is shown. Indeed, in most situations, the nodes will precess at a mostly constant rate.

Example on a Молния orbit with RSS:

Mean apsis altitudes

The ranges of the altitudes of the mean periapsis and mean apoapsis are also shown.

These are not the ranges of the actual lowest altitudes and highest altitudes reached over the course of an orbit; instead those are the ranges of the quantities $h_{\text{pe}}=(1-e)a-r_{\text{planet}}$ and $h_{\text{ap}}=(1+e)a-r_{\text{planet}}$, where $e$ and $a$ are the mean elements above. They can be useful to get an intuition of shape and size of the orbit: “a 425 km × 445 km orbit” is a more practical characterization than “an orbit with semimajor axis of 6806 km and eccentricity of 0.00147”.

Important

These numbers should not be relied on to determine whether the orbit risks reentering or running into terrain; instead the Lowest altitude indicator should be used.

Example on an eccentric lunar orbit in RSS:

Graphs

This section is a draft describing a feature which is not yet available. It will be released in Leibniz.

A button next to the words Mean orbital elements toggles the displaying of graphs. When graphs are shown, a graph of the elements over time is shown to the right of the range of that element for each of $a$, $e$, $i$, $Ω$, $ω$, as well as for the altitudes $h_{\text{pe}}$ and $h_{\text{ap}}$ (see Mean elements above).

The horizontal range of these graphs is the analysed period, given at the top of the report, starting from the current time. For $e$, $i$, $Ω$, and $ω$, the vertical range is the range of values of each element given to the left of the graph. For $a$, $h_{\text{pe}}$, and $h_{\text{ap}}$, the vertical range corresponds to the distances attained over the course of the analysed period:

• for $a$, the vertical range is $[\min r_{\text{pe}}, \max r_{\text{ap}}]$;
• for $h_{\text{pe}}$, and $h_{\text{ap}}$, it is $[\min h_{\text{pe}}, \max h_{\text{ap}}]$.

Example on a frozen low lunar orbit in RSS:

In addition, two larger graphs are shown: The locus of the eccentricity vector and of the Лидов parameters over the course of the mission duration.

Eccentricity vector

This graph shows the locus of $(e\ \cos\ ω, e\ \sin\ ω)$ over the course of the analysed period. It can be used to find frozen orbits.

Examples: the eccentricity vector plots of two low polar lunar orbits about the Moon in RSS, one unstable and one frozen:

Лидов parameters

This graph shows, in red, the locus of the Лидов(Lidov) parameters $(c_2, c_1)$ over the course of the analysed period; these parameters are defined as in [Лид61]:

• $c_2 ≔ e^2 (\frac25-\sin^2\ i\ \sin^2\ ω)$;
• $c_1 ≔ (1-e^2) \cos^2\ i$.

It can be used to understand the expected inclination–eccentricity exchange due to third bodies (the von Zeipel–Лидов–古在 mechanism).

Two grids can be shown: the lines of minimum inclination and maximum inclination corresponding to points in $(c_2, c_1)$ space, or the lines of maximum inclination and minimum eccentricity.

Example: on a stable eccentric lunar orbit in RSS.

Orbit stability

Besides describing the general look and feel of the orbit, the elements can be used to determine whether an orbit is stable.

For a stable orbit, the variation of the semimajor axis and inclination will remain small.

Most orbits will experience a precession (long-term drift) of $Ω$ and $ω$ (nodal and apsidal precession).

Example: a medium Earth orbit experiencing both nodal and apsidal precession. The periapsis starts over the southernmost point of the orbit (bottom of the eccentricity vector graph), and over time moves drifts around the orbit with a positive apsidal precession of roughly two degrees per day, to be over the equator on the ascending pass (right), then over the northernmost point of the orbit (top), over the equator on the descending pass (left), coming back south.

When orbiting an oblate body, the nodal precession can only be eliminated if the orbit is strictly polar.

Example: a low polar Earth orbit. The longitude of the ascending node does not drift, there is no nodal precession. There is a negative apsidal precession, accompanied by a variation in eccentricity.

Frozen orbits about an oblate body

It is often interesting to eliminate the apsidal precession and the variation of the eccentricity, as this means that the satellite will fly over the same latitudes at the same height. This is useful for communications satellites on eccentric orbits, such as Молния, in order to keep the apoapsis (where the satellite spends the most time) above the latitudes being serviced. It is also useful for low Earth orbit observation satellites, as it means that the same locations will be imaged in the same conditions: the satellite being at the same altitude, it will have the same field of view.

On a more pragmatic note, curtailing the variation of the eccentricity also means that the periapsis will not change too much, and in particular will not go under the surface: an orbit with constant eccentricity will not wander into the ground over time.

Orbits that have no apsidal precession and roughly constant eccentricity are called frozen.

These can be classified (following the terminology of Ulrich Walter [Wal18]) in type I and type II frozen orbits.

Type I frozen orbits: the critical inclination.

If the inclination is $i = 63\rlap{°}.43$, the critical inclination, apsidal precession due to the geopotential is eliminated. This means that $ω$ will remain constant, i.e., the apoapsis will remain in the same place on the orbit, regardless of the eccentricity and semimajor axis (note that, because of higher degree terms, the exact value of the critical inclination actually depends slightly on the semimajor axis).

This property is used by the orbit of the Молния satellites, in order to keep the apoapsis over the northern latitudes that they service.

Example: a Молния orbit. The argument of perigee remains around 270°, so that the apogee remains at the northernmost point of the orbit.

Note

A frozen orbit is not necessary to use eccentric orbits that require stability in the location of their apsides. QZSS uses a geosynchronous eccentric orbit with an argument of perigee $ω=270°$ and an inclination in exccess of 40°, so that the apogee can be over Japan, but the apsidal precession is low enough ($6\rlap{°}.6$ per year) that it is viable.

Type II frozen orbits: frozen eccentricity.

The other kind of frozen orbit requires a frozen argument of periapsis $ω = 90°$ or $ω = 270°$, i.e., apsides over the poles. In that case, the eccentricity must have a specific value, the frozen eccentricity, which depends on the inclination and the semimajor axis, as well as the gravitational field of the primary body.

Note

The dependence of the frozen eccentricity on the inclination and semimajor axis can be complex; we recommend studying the literature on the subject, and, in practice, experimenting to see which changes to the orbit reduce the drift of the pattern of the eccentricity vector until it is periodic.

In the case of the Earth, with $J_3<0$, these orbits must have $ω = 90°$ (periapsis over the North pole), as shown in [Cap12, p. 497]. In the case of the Moon, whether a frozen orbit exists for $ω = 90°$ or $ω = 270°$ depends on inclination and semimajor axis, see [Lar11, p. 193]

Both the frozen argument of periapsis and the frozen eccentricity are mean values over the ground track cycle; the mean values provided by the orbit analyser are over a single revolution, and will vary over the course of a ground track cycle. However, in these frozen orbits, the eccentricity vector will trace a periodic path whose period will be that of the ground track cycle; this can be used to find and refine a frozen orbit.

Example 1: a frozen orbit about the Earth.

Example 2: a frozen orbit about the Moon (Orbit C from [RL06]).

Example 3: an unstable (not frozen) low polar orbit about the Moon. The eccentricity vector traces a pattern which does not repeat, but instead drifts away.

The von Zeipel–Лидов–古在 mechanism

The orbit of satellite under the influence of a third body orbiting roughly in the equatorial plane, such as a satellite of the Moon perturbed by the Earth, a satellite of Mercury perturbed by the Sun, etc., experiences a periodic exchange between its inclination and its eccentricity: the eccentricity increases as the inclination decreases, up to a maximum of eccentricity and minimum of inclination, and vice-versa. This is known as the von Zeipel–Лидов(Lidov)–古(Ko)在(zai) mechanism. The Лидов parameters can be used to describe this mechanism. When the orbit is only perturbed by a third body in the equatorial plane of the primary, these parameters remain constant.

Note

In the following, the statements about the behaviour of an orbit are all made under the assumption that there are no perturbations other than the third body. In addition, the orbits are taken to be prograde ($i<90°$). For retrograde orbit, the statements about $i$ below apply to its complement $180°-i$ instead (thus $i≥\mathrm{arc} \cos\sqrt{3/5}≈39\rlap{°}.23$ for prograde orbits corresponds to $i≤180°-\mathrm{arc} \cos\sqrt{3/5}≈140\rlap{°}.77$ for retrograde orbits.

The space of Лидов parameters consists of two regions, separated by a the vertical axis $c_2=0$ on the graph. Adding to the graph the same letters used by [Лид61], we have a triangular region $ABO$ on the right, where $c_2>0$, and a region $DOE$ where $c_2<0$:

For an orbit whose parameters are on the left, with $c_2<0$, the argument of periapsis $ω$ oscillates about a fixed value. For an orbit on the right, with $c_2>0$, the argument of periapsis precesses.

The value of the Лидов parameters determines the amplitude of the eccentricity–inclination exchange; given values of $c_2$ and $c_1$ (i.e., given a point on the graph), the maximal eccentricity $e_{\text{max}}$ and minimal inclination $i_{\text{min}}$ (one extreme of the cycle), as well as the minimal eccentricity $e_{\text{min}}$ and maximal inclination $i_{\text{max}}$ (the other extreme) are known. These can be read off the graph by showing the corresponding grids.

These lines also allow us to identify some important parts of the graph:

Name	Description	Region
$AB$	Both the $i_{\text{min}}=0$ and $i_{\text{max}}=0$ lines form the boundary on the right of the $c_2>0$ region, thus equatorial orbits lie on that line.
$EB$	The $i_{\text{max}}=90°$ line is the bottom of the graph, thus polar orbits lie on that line; further, this is also the $e_{\text{max}}=1$ line, there is no limit to the increase in eccentricity of a polar orbit—it will eventually encounter the ground.
$DE$	On the curved boundary on the left of the graph, $e_{\text{min}}=e_{\text{max}}$ and $i_{\text{max}}=i_{\text{max}}$, thus orbits with these Лидов parameters do not experience any eccentricity–inclination exchange. These orbits all have $i≥\mathrm{arc} \cos\sqrt{3/5}≈39\rlap{°}.23$.
$AO$	The vertical line $c_2=0$ corresponds to $e_{\text{min}}=0$: all circular orbits lie here.
$AD$	The top part of the $c_2=0$, $e_{\text{min}}=0$ line also has $e_{\text{max}}=0$ and $i_{\text{min}}=i{\text{max}}$: these are circular orbits with $i≤\mathrm{arc} \cos\sqrt{3/5}≈39\rlap{°}.23$. They do not experience any eccentricity–inclination exchange. (In contrast, any circular orbit with a larger inclination is subject to eccentricity–inclination exchange, and thus does not remain circular.)

Tip

In practice, when orbiting an oblate body at low altitudes, the effects of the gravitational field of the primary average out those of the von Zeipel– Лидов–古在 mechanism. Low polar orbits can therefore be stable, as shown in the section on Type II frozen orbits.

Caution

When using the stock KSP solar system, there is no oblateness, so the von Zeipel–Лидов–古在 mechanism does not get averaged out; all polar orbits, no matter how low, eventually crash.

Note

When the perturbing body does not lie in the equatorial plane, when the primary has a complex gravitational field, or when there are additional perturbations, the Лидов parameters do not remain exactly constant; however, so long as they roughly remain in the same area of the graph, they can indicate the long-term impact of third-body effects.

Example 1: a stable eccentric lunar orbit, Orbit B from [RL06]; the parameters lie on $DE$, so this orbit does not experience eccentricity-inclination exchange under the influence of the Earth; it can be expected to remain at an inclination of 50° and an eccentricity of 0.5.

Example 2: A highly inclined orbit about Mercury with an initial eccentricity of 0.5. It is doomed.

Example 3: An orbit about Venus with an initial eccentricity of 0.5 and an initial inclination of 45°. It experiences a periodic eccentricity-inclination exchange with apsidal precession.

Example 4: Two high lunar orbits, both exhibiting a periodic eccentricity-inclination exchange. The first one has $c_2<0$ and has no apsidal precession, the second has $c_2>0$ and exhibits apsidal precession.

$c_2<0$	$c_2>0$

Recurrence properties

The treatment of that subject in the orbit analysis tooling, and in particular the notation of the recurrence triple, is heavily inspired by Michel Capderou (2012), Satellites : de Kepler au GPS [Cap12]. The figures in this section are produced using Capderou's tool Ἰξίων.

Background on ground track recurrence

In many cases, a satellite orbit is designed to have a repeating ground track, i.e., to be periodic in the surface frame.

The most obvious example of this situation is a geosynchronous orbit, whose orbit traces a single closed loop on the surface of the Earth, or, in the geostationary case, a single point on the equator.

Semisynchronous orbits, e.g., Молния or GPS, are also an example; these make two nodal revolutions before the ground track returns to its starting point.

Молния	GPS

Many satellites however have more complex recurrence relations; for instance the ground track of ГЛОНАСС satellites repeats every 17 revolutions, which corresponds to 8 days; the ground track of Galileo satellites repeats every 17 revolutions, which corresponds to 10 days.

Low Earth Orbit satellites also tend to have a ground track recurrence, especially when they are dedicated to Earth observation. For instance, the ground track of Sentinel-3A repeats every 385 revolutions, which corresponds to 27 days. A whole-Earth map of this ground track over the whole cycle would be unreadable: it traces a tight net around the planet. We show instead the whole-Earth ground track over a couple of days, and the ground track over Western Europe over one ground track cycle.

The tool

The orbital analysis tool attempts to determine a cycle which is a good match to the current orbit. While this works reasonably well once the satellite is on the intended orbit, the tool cannot infer your intentions; if you would like to use it to align your orbit to a particular recurrence cycle, you should uncheck the Auto-detect box, and type in the appropriate numbers of revolutions and days per cycle.

The recurrence triple

The recurrence triple consists of three numbers $[ν_o; D_{T_o}; C_{T_o}]$.

$C_{T_o}$ is the number of days per cycle.

$ν_o$ is the number of orbits per day, rounded to the nearest integer.

$D_{T_o}$ is the number of orbits per cycle in excess of $ν_o$ $C_{T_o}$. In other words, the number of orbits per day is $ν_o + \frac{D_{T_o}}{C_{T_o}}$.

This means that orbits that have similar periods will have similar values of $ν_o$, the approximate number of orbits in a single day: even though а ГЛОНАСС(GLONASS) satellite has an 8-day, 17-revolution cycle, a Galileo satellite has a 10-day, 17-revolution cycle, a GPS satellite has a 2-day, 1-revolution cycle, and a 北(Běi)斗(Dǒu) MEO satellite has a 7-day, 13-revolution cycle, all have $ν_o = 2$, i.e., they are close to semisynchronous.

If $D_{T_o} = 0$, there is an integer number of orbits per day; the orbit is synchronous or sub-synchronous. Note that this induces stronger resonances with the geopotential, so such orbits are less stable than similar orbits with a longer cycle.

The equatorial shift, grid interval, and subcycle

The equatorial shift is the angle, measured on the surface, between successive crossings of the equator; it measures how much the orbit shifts from one ascending pass to the next (as seen from an observer on the ground). The corresponding distance along the equator is given in parentheses. On planets with prograde rotation, the equatorial shift is almost always negative, i.e., westwards.

In this map, we can see an equatorial shift of -25°.2, or 2810 km: the ground track starts north of New Guinea, and, one revolution later, is 2810 km west of its starting point, over Borneo.

The grid interval is the angle, measured on the surface, between neighbouring tracks on the equator; note that neighbouring tracks may be many revolutions apart. As for the equatorial shift, the corresponding distance along the equator is given in parentheses.

In this map, we can see a grid interval of 0°.93, or 104 km: the ground track traces a grid whose spacing on the equator is 104 km; over the course of 27 days, the satellite will fly over locations 104 km apart along the equator.

The grid interval is important for observation satellites: the instrument swath (the extent of the surface which is seen by the instrument) must be wider than the grid interval in order for the whole surface to be observed.

The subcycle is the number of days after which the ground track comes back to within one grid interval of its starting point.

In this map, we can see the ground track of Sentinel-3A over one subcycle (4 days); the last ascending pass (in blue) is one grid interval away from the first one (in red), north of New Guinea. All ascending passes before that are further away.

The subcycle measures how quickly the satellite will come “close” (as defined by the grid interval) to any point on the surface; a short subcycle allows the satellite to quickly observe any given target; a long subcycle may require waiting a full ground track cycle before an observation can be made.

Equatorial crossings

The properties of the recurrence described in the preceding section are purely theoretical. The analysis of the equatorial crossings of the actual orbit describes how well that orbit follows the nominal recurrence.

The passes are numbered as follows: on the recurrence grid, the ascending pass whose equatorial crossing has the smallest longitude East of the reference meridian is pass 1; subsequent passes are numbered chronologically, with revolution n comprising ascending pass 2n-1 and descending pass 2n: odd passes are ascending, even passes descending.

The longitude of ascending pass differs from the previous one by the equatorial shift.

The longitudes of the equatorial crossings of the ascending and descending passes are given as ranges of longitudes on the equator. The midpoint describes where the recurrence grid is situated on the planet; the width describes how much the satellite deviates from the nominal recurrence grid. This deviation is also indicated as a distance measured along the equator.

On a well-kept orbit, this deviation should be much smaller than the grid interval.

Mean solar time

(Not yet documented)