Rocket Mechanics From Kerbal Space Program Part 4
- mollievonehr
- May 14
- 14 min read
by Michael Lingg
Inner-System Transfers
Transfer to another body in system
In the previous part (Part 3) we looked at how a spacecraft would travel to and from a moon, or transfers between two bodies when one body is orbiting the other.
Inner-system transfers cover flying between two bodies orbiting the same parent body, such as travelling from Kerbin (Earth analog) to Duna (Mars analog) which are both orbiting Kerbol (Sun analog). This type of transfer can be done using the same maneuvers we have already looked at when transferring to and from a moon, further when we know how to apply these concepts to inner-system transfer, we can further extend the concepts to transfers between two bodies of any configuration.
Simple Transfers
First let's look at the simplest method to transfer to another planetary body in the system. In the KSP base game, Kerbals have no life support needs, live forever (or at least for as long as the game can simulate), and there is time acceleration. As a result, we have the option to break inner-system transfers up into independent stages, simplifying planning even if the result will not be efficient. As previously mentioned we will use Kerbin as our origin or departure body, and Duna as our destination.
The first stage is getting into orbit around Kerbin, if you aren’t already, this was covered in part 1 (Part 1).
The second stage is moving from an orbit around Kerbin to an orbit around Kerbol. In a simple KSP transfer I do this by exiting the SOI of Kerbin (see part 3 for a discussion of the SOI Part 3). This part is rather simple by itself, burn prograde (see part 2 for maneuvering discussions Part 2) until the spacecraft’s path will exit Kerbin’s SOI, in other words we want to fly far enough away from Kerbin so that Kerbin’s gravity has minimal impact on the spacecraft’s orbit around Kerbol. In KSP this is simple as the simulation shows you when your flight path will leave the influence of Kerbin as shown below. In the real world a spacecraft will always be gravitationally affected by all bodies, and even minimal effects can make a significant difference in space travel, so the computations become more complex. Now we wait until the spacecraft leaves Kerbin’s SOI before the next stage. This can take 1-2 days in KSP, and 3-5 days in the real world. An interesting note is that the moon is a bit under halfway to the edge of Earth’s SOI, so Apollo moon landings were half way to solar orbit.
The image below shows the path of a spacecraft leaving Kerbin’s SOI on the left as the orbit is an open hyperbola rather than a closed ellipse. The right side shows the orbit of the spacecraft around Kerbol in dark blue, as opposed to the orbit of Kerbin, which our spacecraft escaped, in cyan.
Now our spacecraft has left Kerbin’s SOI and we are orbiting Kerbol and ready to travel to Duna. Conceptually this is no different from orbiting Kerbin and wanting to fly to the Mun, just that the distances are much higher. We can use the same computations described in part 3 to identify the phase angle when we need to burn to match orbits with Duna. Since our basic Kerbals can be very patient and not worry about snacks, we could be in orbit around Kerbol far away from our transfer burn point as shown in the image below. In the image the spacecraft is at the top of the orbit, but must wait until it reaches the bottom of the orbit for a proper Hohmann transfer to Duna. This does not cost us much in fuel as we are just orbiting Kerbol until it is time to transfer, all we are really wasting is time. In the next section we will discuss methods to reduce this time cost.
While burning to Duna will be a prograde burn to increase our orbit to match Duna, transferring to a planet like Eve (Venus analog) uses the same computations for the phase angle, we just need to burn retrograde to reduce our orbit to match Eve. Once the burn is complete, we again have to wait, even longer this time as transferring from Krebin to Duna can take about 214 Kerbin days (A Kerbin day is 6 hours long, so this is ~53 Earth days), and transferring from Earth to Mars can take about 259 Earth days. This can vary by 20-30% due to orbital variations and transfer methods.
Once our spacecraft reaches Duna’s SOI, entering orbit is no different from what we discussed for transferring to a moon, and returning to Kerbin uses the same process we just described.
More Efficient Transfers
There are a number of ways that the transfer can be made more efficient in terms of fuel or time, though these requirements conflict somewhat. We can change the timing of the launch, the timing of the transfer to Kerbol orbit, or the type of transfer orbit used. We will discuss a few of these methods here and some of the tradeoffs.
Launch at the Right Time
This is a rather basic optimization which you could easily skip reading but is here to make the reader think a little. In my example I do not specify when to launch to orbit, the rocket is just launched willy nilly. Since Kerbals don’t care about time and supplies, they can just drift until it is time to transfer to the destination body. In the real world, we would prefer not to have our astronauts just floating around in space indefinitely, they tend to like to breathe and eat, so we typically want to reduce their time in space. We can do this by launching when the spacecraft will reach Kerbol orbit at the time of the transfer window. So we subtract the amount of time it takes for the spacecraft to reach the transfer window from the time when we want to perform the transfer burn, then launch from the ground at that time.
Like I said, pretty basic optimization.
Direct Transfer
So this comes with a couple of different flavors. First we will look at direct transfer from low Kerbin orbit.
Direct Transfer from Low Kerbin Orbit
This improvement builds a little bit on launching at the right time. Not only do we launch at the right time, but we try to perform a single burn from low Kerbin orbit to create the Kerbol transfer orbit to intersect with Duna. This requires not only making the burn at the right time to we properly transfer from Kerbin to Duna, but also burning at the right angle around Kerbin so we can continue burning after we have enough speed to leave Kerbin’s orbit, and we will continue shifting our Kerbol orbit to an intercept with Duna, while we are still burning in low Kerbin orbit. The image below shows on the left side how our spacecraft’s orbit in purple is only a little offset from Kerbin’s orbit in cyan when the spacecraft is moving just fast enough to exist Kerbin’s SOI. If our spacecraft instead kept burning, while at the right point in Kerbin’s orbit, the spacecrafts future orbit around Kerbol would keep increasing the apoapsis until the orbit intersected with Duna.
We discussed in part 3 how to return from an orbit around a moon, and burn at the right point in the orbit in order to lower the upcoming Kerbin orbit the the appropriate periapsis for reentry. The same principles apply for changing the Kerbol orbit while in low Kerbin orbit, though the direction of orbit (reversed from the Mun return), and if we are increasing or decreasing our next orbit (decreasing during the return from the Mun and increasing to transfer from Kerbin to Duna), has to be considered. The image below shows the optimal time to burn from low Kerbin orbit to lead to an eventual Kerbol orbit that will intersect with Duna.
Direct Transfer from Kerbin Surface
Now that we know where and when we need to burn to transfer to Duna directly from low Kerbin orbit, we can further back track this to the surface and determine where Kerbin needs to be in its day cycle and orbit in order to launch directly from the surface and produce a nonstop burn (well staging will happen) into a direct transfer orbit to Duna. A burn like this would benefit the most from the Oberth effect (where the higher your present speed, the more acceleration you will gain for a given amount of fuel) as well as being the shortest possible (I think) transfer to Duna given a standard Hohman transfer between Kerbin and Mars.
While I think this is the most efficient possible transfer as far as fuel, NASA does not use this approach. The size and complexity of the rocket needed for a direct ascent is significant. NASA also prefers a parking orbit around Earth, and a parking solar orbit before burning for the destination, in order to make sure things are still working properly and the spacecraft is ready for the next stage. Further, real world rockets can tailor stages for launch, leaving Earth’s SOI, performing the solar orbit transfer burn, etc, making the rocket very efficient for each stage. This can somewhat be done in KSP, but since KSP has specific size fuel tanks and rocket engines which we assemble like legos, KSP rockets tend to be overengineered. This makes a direct transfer somewhat more practical within KSP.
Gravity Assists
If we want to use as little fuel as possible, gravity assists can significantly reduce the amount of fuel used for solar orbit maneuvers. Changing orbits costs a lot. If we can pass close enough to a large body in space (any of the planets or even the moon), we can use the large body’s gravity to change our spacecraft’s solar orbit. By passing behind the body our spacecraft is accelerated in solar orbit, by passing ahead of the body our spacecraft is decelerated in solar orbit. While this changes the orbit around the large body like performing a radial inward burn, the change to solar orbit is like a prograde or retrograde burn performed in solar orbit. The image below shows how our spacecraft travelling toward Eve could be accelerated to a higher Kerbol orbit (the purple circle) by passing behind Eve (left), or a lower Kerbol orbit by passing ahead of Eve (right).
Nearly all of the real world probes launched into solar orbit have used radial burns in one way or another. Voyager was significantly accelerated by Jupiter, more than doubling its speed to help send the probe rocketing away from the Sun. The Parker Solar Probe was launched with a solid rocket booster to propel the probe toward a low perihelion around the Sun, but the mission wanted to pass closer to the Sun yet. The probe used small onboard thrusters to adjust its orbit to pass ahead of Venus at the right angle to slow the spacecraft and lower its orbit to be closer to the Sun. This gravity assist was performed a total of seven times to send the probe closer to the sun than any previous spacecraft.
Fast Transfers
In this improvement we are not looking to optimize fuel use, but to instead trade fuel for time. A Hohman transfer from one planet to another results in the lowest velocity difference between the spacecraft and the destination body, but this is not necessarily the fastest transfer. There are two definitions of a fast transfer. The first is the fastest time from the transfer burn to intercepting the target body, however this transfer requires the proper orientation between the origin and the destination. The second definition of a fast transfer is the fastest transfer if we launch and perform the transfer burn right now.
Fastest Possible Transfer
The fastest transfer between two orbital bodies is more about how much delta v can you get out of your rocket (total thrust over time), rather than specific orbital mechanics. In a Hohman transfer you will intercept the target body at the apoapsis from the transfer burn, 180 degrees away from the starting point. As you burn more than needed for the Hohmann transfer, the point where your spacecraft intersects the destination body’s orbit moves backward along the destination body’s orbit until at infinite speed it would simply be a transition between the closest positions of the two bodies. However we don’t have infinite fuel or acceleration. So the fastest possible transition is limited by using about half the delta v to accelerate the rocket toward the target body, and about half the fuel to match velocities with the target body, with (I believe) the matching burn usually being slightly lower due to gravity effects. Then the fastest time is reduced further if we also have to save fuel to return back to our body of origin.
The image below shows how increasing the speed past the base Hohman transfer delta v can lead to a closer interception point. The leftmost maneuver is a near direct transfer from Kerbin to Duna, the middle image shows a more typical fast transfer, while the rightmost image shows a standard Hohmann transfer. The increased transfer speed comes at a cost, a Hohmann transfer is about 1100 m/s, while the fast transfer in the middle is about three times faster at 3225 m/s, and the direct transfer would require over an additional four times more at 14000 m/s.
Fastest Transfer Now
Next we look at how much we can reduce the time to intercept the target body if we leave right now. While this transfer can be continuously reduced by higher rocket velocities (eventually simply a vector from the origin to the destination, pushing through the sun’s gravitational effect), we will focus on the fastest transfer within normal orbital mechanics.
The fastest transfer using a normal orbital transfer for any given departure time depends on the relative position between the origin and destination bodies. If they are in an ideal position for a Hohman transfer, the Hohmann transfer will be the fastest, barring some chance, complicated harmonic transfer. If we burn prograde longer to reach the destination’s orbit faster, we will get there before the destination. This is illustrated in the image below where the spacecraft intersects Duna’s orbit a little above the rightmost point, and the marker pointing up shows where Duna is slightly ahead in its orbit where our spacecraft will pass at the marker pointing down.
However if our destination is not in the proper orientation for a Hohmann transfer, and we are willing to burn the fuel, we do not have to wait for a Hohmann transfer. If the destination is behind the orientation for a Hohmann transfer the spacecraft can burn for a higher orbit than the destination, letting the destination catch up. Then the spacecraft can burn again to return to the orbit of the destination, intersecting with the destination sooner than waiting for a Hohmann transfer. If the destination is ahead of the ideal orientation for a Hohmann transfer, the reverse can be performed where the spacecraft burns for a lower orbit to catch up with the destination before burning for an interception. This is illustrated in the image below where the first burn is performed at the bottom point of Kerbin’s orbit, leading to the transfer with the yellow dotted curve. A second burn is performed as the spacecraft passes beyond Duna’s orbit in the upper right, leading to an interception with Duna, resulting in the reddish dotted orbit which shows what our spacecraft’s final orbit would be, if it continued past Duna.
Spiral Transfers
Spiral transfers are not a standard type of orbital transfer, but one created by the type of engine used. Very low thrust engines, such as ion engines, will burn for a very long time to produce a transfer. A Hohmann transfer is a short high power thrust, resulting in an elliptical orbit. Ion engines on the other hand produce a very small amount of thrust. The Dawn probe ion engine was 1,000,000 times less thrust than the Apollo SMS engine. So the Dawn probe is lighter, but even the thrust to weight ratio is 33,000 times smaller on the Dawn probe than the Apollo SMS. Instead an ion engine can run for a very long time. The Dawn probe ran for about 11 years from launch to final engine shutdown, during that time the ion engine was providing thrust for nearly 3 years cumulatively. For this type of orbital maneuver, the spacecraft continues to burn slowly in the direction of the orbit (if the spacecraft wants to increase the apoapsis). The result is the transfer orbit for such a maneuver slowly spirals outward until the orbit intersects with the destination object. Gravity assists can be greatly beneficial for this type of spacecraft. This method is not recommended for human transport, when travelling faster to reduce the need for supplies is a bonus.
Ballistic Capture
In addition to the discussion of transfer orbits, I want to talk about a couple methods used to make orbiting around our destination a little more efficient. Ballistic capture is something we talked a little bit about in the post about transferring to the Moon (or Mun if you are looking at the KSP part). We discussed earlier how gravity assists can be used to increase or decrease orbital speeds, this is also used to help enter orbit around our destination. We talked about how some Apollo missions passed in front of the Moon, using the Moon’s gravity to pull Apollo into an orbit (sometimes very eccentric) around the Moon. This method can be used with any destination body to reduce the fuel costs needed to enter orbit.
Aerobraking
A second method is aerobraking. Here we see a Kerbal rocket flying through the atmosphere to slow the rocket down. As long as we reach an orbit, even if highly eccentric, repeated aerobraking can be used to slow the rocket until it can safely be captured by the atmosphere and land, all without using nearly as much fuel. This is a little too easy to perform in KSP, heating and structural stresses due to this maneuver are lower than in the real world. However this method is still viable with proper planning and a good heat shield and dissipation. The Mars Reconnaissance Orbiter used hundreds of very shallow entries into the atmosphere to slowly reduce the Orbiter’s speed while minimizing the heating and stress on the aircraft.
Delta-V Map
Now that we know how to get to other bodies in a solar system, how do we know if we can get there? To help with this question, I am introducing the Delta V map.
I am starting with the KSP version as I feel it is rather less complex than the Earth solar system Delta-V map. To reach Duna, like in our examples above, we follow the path from Kerbin to Duna. This starts at the bottom where we see a circle representing Kerbin and follow the blue line up to Low Kerbin Orbit of 80km. Next we follow the next blue line up from the 80km orbit to orbit around Kerbol. The red line shows the transfer from Kerbol orbit to a Duna flyby. Finally the remaining three red line segments show the cost to enter orbit around Duna, transition to a low circular orbit around Duna, and finally land on Duna. These segment delta v costs are:
Kerbin -> Low Kerbin Orbit: 3400 m/s
Low Kerbin Orbit -> Kerbol Orbit: 950 m/s
Kerbol Orbit -> Duna Flyby: 150 m/s
Note I believe this is the cost from Low Kerbin Orbit to continue burning to match Duna orbit as we discussed in the direct transfer. If I use the KSP planetary transfer maneuver tools to reach Duna from orbit around Kerbol at the altitude of Kerbin’s orbit, the cost is closer to 1100 m/s
Duna Flyby -> Duna Orbit: 250 m/s
Duna Orbit -> Low Duna Orbit: 350 m/s
Low Duna Orbit -> Duna Surface: 1450 m/s
Note that none of these costs consider ballistic capture or aerobraking.
So the total cost to start from the surface of Kerbin and land on Duna is:
3400 + 950 + 150 + 250 + 350 + 1450 = 6540 m/s
This does not include the costs to return from Duna if this is not a one way trip.
Now let’s look at a delta v map for travelling from Earth to anywhere in the solar system.
Much like the KSP map, Earth’s solar system delta-V map can be read by following a sequence of maneuver costs. Heinlein famously said, "Getting to orbit is halfway to anywhere." While not mathematically precise, the sentiment holds, reaching orbit represents a major portion of the energy cost for interplanetary travel. Looking closely at the solar system delta-V map, you’ll notice that for roughly twice the cost of reaching Low Earth Orbit, you can reach the surface of Mars, orbit Venus or Mars, or perform a flyby of Jupiter or (almost) Saturn.
This is one reason why the ability to source fuel from space, whether mined from the Moon, asteroids, or Mars, could dramatically reduce the need to haul fuel up from Earth, making long-range exploration far more economical. I explore this idea in Book 2 of my ongoing Icarus Program story, where the program researches and funds mining ventures to the Kerbal Space Program moons. This work leads into the upcoming Book 3, which imagines how in-situ resource utilization could revolutionize deep space travel. [Book 1]. (Book 1 chronicles the birth of the Icarus Program and humanity’s first steps into space: [Book 2].)
Kommentare