Landing Site Prediction on Return from Deep Space

[OhioBob]

Over the course of the last couple months I started to develop and use a method to predict the location of my landing site on Kerbin when returning from deep space. I just recently formalized the technique by producing some useful charts and equations. I figured I might as well share it with the rest of the KSP community. Perhaps somebody else here might find it useful.

When a spacecraft is returning from deep space (i.e. interplanetary space) and on course for a direct entry into the Kerbin atmosphere, we have only limited control over where the spacecraft will land. It is nice to have some idea of where we're going to land while still a considerable distance from Kerbin. Being able to predict the landing site lets us know in advance whether we'll be descending into a safe area or a hazardous area. If we know the spacecraft will be descending onto the ocean or a flat plain, then we can relax and not have much to worry about. However, if the prediction shows that we're likely to descend onto the steep slopes of a mountain, then we likely need to alter our trajectory the best we can to avoid potential disaster.

The method presented here allows us to deduce and, to some degree, control the location of the landing site while near the edge of Kerbin space and still days or hours away from entry. The method also works for predicting the landing site for returns from Mun and Minmus.

The numbers that follow were generated by performing computer simulations outside of the game environment. In the instances where I've been able to put the technique to use in actual game conditions, the results have matched the theory pretty closely. The simulations are based on stock aerodynamics. Although the basic concept and procedural steps should be valid with any aerodynamic model, the specific numbers presented are valid only with stock aero.

Step 1 - Compute hyperbolic excess velocity.

To determine the landing site we must know at what speed we're approaching Kerbin. We do this by computing the hyperbolic excess velocity, v_{∞}, of our trajectory within Kerbin's sphere of influence. Shortly after entering Kerbin’s sphere of influence, briefly phase the game and note the spacecraft’s altitude, h, and velocity, v. This can be done at any point along the trajectory, and any pair of h-v values will do, as long as they are taken at the same time. The hyperbolic excess velocity is computed using the following equation:

v_{∞} = [ v² – (2μ/r) ]^1/2

where μ is Kerbin’s gravitational parameter, equal to 3.5316×10¹² m³/s², and r is the radius distance, equal to h + 600,000 meters.

Skip this step for returns from Mun and Minmus (or any elliptical orbit within Kerbin’s SOI).

Step 2 – Determine target periapsis and alter trajectory.

The next step is to establish a trajectory that allows us to predict the flight path of the spacecraft through Kerbin’s atmosphere. The trajectory we choose is that in which the spacecraft’s landing site is directly beneath the periapsis of the orbit. This condition can be achieved by carefully selecting and controlling the periapsis altitude of the trajectory. Periapsis altitude, Pe, as a function of v_{∞} is obtained using the following graph:

The periapsis altitude can also be calculated using one of the following equations:

For v_{∞} ≤ 2000 m/s,

Pe = 0.0000001416 v_{∞}³ – 0.0007332 v_{∞}² + 0.02001 v_{∞} + 21970

For v_{∞} > 2000 m/s,

Pe = 0.00011174 v_{∞}² – 1.7053 v_{∞} + 23174

For a return from Mun, use Pe = 22,380 m, and for a return from Minmus, use Pe = 22,090 m.

Now perform a burn to set the periapsis of the trajectory to the altitude determined above (as closely as can be reasonably done). In general, to raise the periapsis burn radially outward, and to lower the periapsis burn radially inward. You can also use this opportunity to adjust the inclination if necessary. I typically like to make sure my trajectory’s swing around Kerbin is as close to the equatorial plane as possible.

Completing this step means that the spacecraft is now on a trajectory to land on the surface of Kerbin at a point that is directly beneath the periapsis. You may note that the periapsis moves once the spacecraft enters the atmosphere and starts to experience drag forces. Ignore this; the landing site will be beneath the original periapsis before it was altered by reentry.

Step 3 – Determine landing site location.

Go to the map view and note the Kerbin terrain that is located directly beneath the periapsis. Since Kerbin rotates, this is not the terrain that will be beneath the periapsis at the time the spacecraft arrives. To determine this we must also note the time to periapsis.

In the time to periapsis we can ignore the days, because for every integer day that passes, Kerbin will return to the exact same position. Simply note the hours and minutes. Since Kerbin’s rotational period is six hours, the planet’s surface rotates eastward 60^o every hour and 1^o every minute. This means that the landing site will be west of the periapsis by the amount that Kerbin will rotate in the time between the present and the arrival of the spacecraft.

To visually identify the landing site, use the following map. Each white vertical line on the scale is separated by a distance of 15^o, or 15 minutes of time. The distance between the slightly longer lines is 60^o, or 1 hour.

Identify on the map the current location of the periapsis. From that position, count off to the west (left) the distance represented by the hours and minutes to periapsis. If you reach the left edge of the map, move all the way to the right and count off the remaining distance starting from the right hand edge. The new position found after counting off the appropriate distance will be where the spacecraft will land.

Step 4 (optional) – Hazard avoidance and trajectory alteration.

If Step 3 reveals the spacecraft is likely to land in a hazardous location, there is only so much that can be done. The landing location can be altered to some degree by raising or lowering the periapsis altitude. Raising the periapsis (decreasing the entry angle) will cause the spacecraft to land long (farther east, assuming a prograde entry). Lowering the periapsis (increasing the entry angle) will cause the spacecraft to land short (farther west). There are limitations, however. If the periapsis is raised too much, the spacecraft will pass through the atmosphere and back into space. If lowered too much, the craft will experience suicidal g-forces (if that even matters to a Kerbal).

By simulation I have determined how much the periapsis altitude must be adjusted to land –15^o, +15^o and +30^o from the landing position found in Step 3. The following table gives the vertical distances that the periapsis should be lowered to land 15^o short, and raised to land 15^o or 30^o long. These distances are a function of v_{∞}. Please be advised that this is only theoretical; I just recently produced the table and haven’t had the opportunity to test the results in game situations.

[TABLE=class: outer_border, width: 500]

[TR]

[TD=align: center]v_{∞}

(m/s)[/TD]

[TD=align: center]Short

-15^o

(m)[/TD]

[TD=align: center]Long

+15^o

(m)[/TD]

[TD=align: center]Long

+30^o

(m)[/TD]

[/TR]

[TR]

[TD=align: center]0[/TD]

[TD=align: center]-20,130[/TD]

[TD=align: center]+7,535[/TD]

[TD=align: center]+9,735[/TD]

[/TR]

[TR]

[TD=align: center]500[/TD]

[TD=align: center]-20,380[/TD]

[TD=align: center]+7,535[/TD]

[TD=align: center]+9,705[/TD]

[/TR]

[TR]

[TD=align: center]1,000[/TD]

[TD=align: center]-21,040[/TD]

[TD=align: center]+7,535[/TD]

[TD=align: center]+9,605[/TD]

[/TR]

[TR]

[TD=align: center]1,500[/TD]

[TD=align: center]-21,950[/TD]

[TD=align: center]+7,520[/TD]

[TD=align: center]+9,465[/TD]

[/TR]

[TR]

[TD=align: center]2,000[/TD]

[TD=align: center]-22,940[/TD]

[TD=align: center]+7,480[/TD]

[TD=align: center]+9,280[/TD]

[/TR]

[TR]

[TD=align: center]2,500[/TD]

[TD=align: center]-23,900[/TD]

[TD=align: center]+7,415[/TD]

[TD=align: center]+9,085[/TD]

[/TR]

[TR]

[TD=align: center]3,000[/TD]

[TD=align: center]-24,740[/TD]

[TD=align: center]+7,330[/TD]

[TD=align: center]+8,885[/TD]

[/TR]

[TR]

[TD=align: center]3,500[/TD]

[TD=align: center]-25,460[/TD]

[TD=align: center]+7,240[/TD]

[TD=align: center]+8,695[/TD]

[/TR]

[TR]

[TD=align: center]4,000[/TD]

[TD=align: center]-26,060[/TD]

[TD=align: center]+7,150[/TD]

[TD=align: center]+8,515[/TD]

[/TR]

[TR]

[TD=align: center]4,500[/TD]

[TD=align: center]-26,530[/TD]

[TD=align: center]+7,055[/TD]

[TD=align: center]+8,345[/TD]

[/TR]

[TR]

[TD=align: center]5,000[/TD]

[TD=align: center]-26,920[/TD]

[TD=align: center]+6,965[/TD]

[TD=align: center]+8,195[/TD]

[/TR]

[/TABLE]

Note that the farther we land long of the position found in Step 3, the more sensitive the final landing position is to small changes in periapsis altitude. Furthermore, unpredictability due to variations in the drag coefficient is magnified when the spacecraft follows a longer path through the atmosphere. We should, therefore, expect greater error in the predicted landing position when attempting to land long. Also note that for the -15^o scenario, Pe becomes negative for v_{∞} greater than about 1,135 m/s. In this case the periapsis is below ground level and will not be visible in the map view.

Given the above, we have what amounts to a 45^o long landing footprint (about 500 km). It should be possible to find a suitable landing site within the available footprint.

For returns from Mun and Minmus we have much greater control over the landing site. This is because we can always delay our departure until Kerbin has rotated into a suitable position, thereby allowing us to place our periapsis over whatever terrain we desire. In this way it is possible to obtain landings very close to Kerbal Space Center.

Acceleration Loads

In the stock game, g-forces are not a concern because there are no detrimental effects from high acceleration loads (or heating). Nonetheless, below I provide the peak acceleration loads that will occur in each scenario. Note that for very high values of v_{∞} the g-forces are extreme. If you are playing a modded form of the game in which acceleration loads matter, then please take notice and use the necessary caution.

In the +30^o scenario we have a simple form of a "skip" reentry. That is, the spacecraft descends into the atmosphere and then, for a brief period, begins to rise in altitude before making its final descent. This results in two peaks in the acceleration curve (the larger is given).

[TABLE=class: outer_border, width: 500]

[TR]

[TD=align: center]v_{∞}

(m/s)[/TD]

[TD=align: center]Short

-15^o

(g)[/TD]

[TD=align: center]Norm

0^o

(g)[/TD]

[TD=align: center]Long

+15^o

(g)[/TD]

[TD=align: center]Long

+30^o

(g)[/TD]

[/TR]

[TR]

[TD=align: center]0[/TD]

[TD=align: center]8.1[/TD]

[TD=align: center]4.2[/TD]

[TD=align: center]2.2[/TD]

[TD=align: center]2.4[/TD]

[/TR]

[TR]

[TD=align: center]500[/TD]

[TD=align: center]8.4[/TD]

[TD=align: center]4.4[/TD]

[TD=align: center]2.2[/TD]

[TD=align: center]2.4[/TD]

[/TR]

[TR]

[TD=align: center]1,000[/TD]

[TD=align: center]9.3[/TD]

[TD=align: center]4.8[/TD]

[TD=align: center]2.4[/TD]

[TD=align: center]2.4[/TD]

[/TR]

[TR]

[TD=align: center]1,500[/TD]

[TD=align: center]10.9[/TD]

[TD=align: center]5.5[/TD]

[TD=align: center]2.8[/TD]

[TD=align: center]2.4[/TD]

[/TR]

[TR]

[TD=align: center]2,000[/TD]

[TD=align: center]13.0[/TD]

[TD=align: center]6.6[/TD]

[TD=align: center]3.5[/TD]

[TD=align: center]2.8[/TD]

[/TR]

[TR]

[TD=align: center]2,500[/TD]

[TD=align: center]15.8[/TD]

[TD=align: center]7.9[/TD]

[TD=align: center]4.3[/TD]

[TD=align: center]3.5[/TD]

[/TR]

[TR]

[TD=align: center]3,000[/TD]

[TD=align: center]19.1[/TD]

[TD=align: center]9.6[/TD]

[TD=align: center]5.4[/TD]

[TD=align: center]4.5[/TD]

[/TR]

[TR]

[TD=align: center]3,500[/TD]

[TD=align: center]23.1[/TD]

[TD=align: center]11.6[/TD]

[TD=align: center]6.7[/TD]

[TD=align: center]5.7[/TD]

[/TR]

[TR]

[TD=align: center]4,000[/TD]

[TD=align: center]27.7[/TD]

[TD=align: center]13.9[/TD]

[TD=align: center]8.3[/TD]

[TD=align: center]7.2[/TD]

[/TR]

[TR]

[TD=align: center]4,500[/TD]

[TD=align: center]33.0[/TD]

[TD=align: center]16.5[/TD]

[TD=align: center]10.1[/TD]

[TD=align: center]8.8[/TD]

[/TR]

[TR]

[TD=align: center]5,000[/TD]

[TD=align: center]38.7[/TD]

[TD=align: center]19.5[/TD]

[TD=align: center]12.2[/TD]

[TD=align: center]10.8[/TD]

[/TR]

[/TABLE]

EXAMPLE

Let’s say we are on a return from Duna. After passing into Kerbin’s SOI, we note that the altitude is 83,400,000 m and the velocity is 920 m/s. We compute v_{∞} as follows:

v_{∞} = [ 920² – (2 × 3.5316×10¹² / (83,400,000 + 600,000)) ]^1/2 = 873 m/s

We next compute the periapsis altitude of the desired trajectory (or read Pe from the graph),

Pe = 0.0000001416 × 873³ – 0.0007332 × 873² + 0.02001 × 873 + 21970 = 21,520 m

We now perform a course correction burn to change our periapsis altitude to 21,520 m.

We go to the game map view and identify the terrain features beneath the current periapsis, and note that the time to periapsis is, say, 3 days, 4 hours, 45 minutes.

We identify on the following map the current periapsis location and count off 4:45 (285^o) to the west to find the projected landing site.

Edited April 9, 2015 by OhioBob

[Specialist290]

Impressive work. Definitely one for the Drawing Board

Incidentally, about how much would you have to modify to apply the procedure to any body with an atmosphere, not just Kerbin? I'd imagine it would be quite useful for people looking to establish permanent bases on bodies like Laythe or Duna (or, for the especially ambitious, Eve) to be able to plot their landings within a reasonable degree of accuracy.

[OhioBob]

Incidentally, about how much would you have to modify to apply the procedure to any body with an atmosphere, not just Kerbin? I'd imagine it would be quite useful for people looking to establish permanent bases on bodies like Laythe or Duna (or, for the especially ambitious, Eve) to be able to plot their landings within a reasonable degree of accuracy.

That probably wouldn't be too hard to do. I already have the computer simulations to figure it out, it's just a matter of putting in the time. Personally, I've never done a direct entry from deep space on any planet other than Kerbin. Every time I've landed elsewhere, I've always entered orbit first. For precise landings from orbit, somebody else has already figured that out. Before I put any further effort into it, I think it would be wise to wait for the revised aerodynamics in 1.0. Anything I do now will soon become obsolete.

[el_coyoto]

Wow, that's a lot of work you've put into that!

I'll have to test this when I restart a new career (yeah, I tend to use MJ landing predictions A LOT ).

I really like the fact that it works for hyperbolic trajectories, while the only other thread I know on the subject deals with circular orbits.

I also enjoyed your explanation on the hyperbolic excess velocity : it's clear, concise and made everything else click in my head.

Also, I've always had trouble taking into account the effect of the rotation of Kerbin of the landing site : I had understood it superficially ("sooo, if Kerbin rotates every 6 hours, and my PE time is X, then it will move my landing site to... Ah dangit, I'm no Scott Manley, I'll just wing it!") but your map helped me grok the concept and map it to something tangible that I can rely on (I'm a very graphic oriented person when it comes to learning and using that knowledge).

Awesome job!

+rep

[OhioBob]

I really like the fact that it works for hyperbolic trajectories, while the only other thread I know on the subject deals with circular orbits.

I use those charts for circular orbits, they work nicely. They were part of my inspiration to develop a similar technique from hyperbolic orbits.

Also, I've always had trouble taking into account the effect of the rotation of Kerbin of the landing site : I had understood it superficially ("sooo, if Kerbin rotates every 6 hours, and my PE time is X, then it will move my landing site to... Ah dangit, I'm no Scott Manley, I'll just wing it!") but your map helped me grok the concept and map it to something tangible that I can rely on (I'm a very graphic oriented person when it comes to learning and using that knowledge).

The map with the scale on it comes in really handy. I keep a printout of it nearby and refer to it every time I'm doing a reentry from Mun, Minmus, or interplanetary space. The fact that it is graduated in 15 minute increments makes it really easy to use.

[OhioBob]

Regrettably I've found an error in one of my formulae used to derive the periapsis altitudes. I've corrected it and have revised all the numbers in the opening post. These new numbers should produce a more accurate landing site prediction. I apologize to anyone who may have used the old numbers.