How do I plan my interplanetary transfer to an inclined orbit?

[Actually_New_KSP_Player]

I've been using transfer window planner for quite a while, but now I just want to do it myself. Most of the guides that I've seen taking Duna as an example, but it's orbit has almost 0 inclination while other planets like Eeloo, Dres and Moho are highly inclined which makes them impossible to get an encounter the "wait for a certain degree between Kerbin, Sun and planet, make a prograde burn, make a mid-course correction if needed" way. The mid-course inclination correction also doesn't seem to be a great option here because the AN and DN nodes often located very close to my Periapsis and may require more than a 1000 of dV to get an encounter.

[Aeroboi]

To Moho and Dress I always use gravity assists.

You can use Duna gravity assist to help the plane change to get to dress, at least partially unless your close to An/Dn in which case Duna can plane change enough. So if you take the transfer planner and Duna is in line with your Dress orbit transfer trajectory you can take Duna as target to help you get there. Personally I always use a combination of Dawn ION and LN-N to make sure I have enough Dv.
I often make SSTA's with a ISRU and a few more EC points past 30 (required for ISRU) makes enough EC for 4 Dawn engines at full throttle so I try to make use of that Dv gain since the coupled ISP of one LV-N and 4 Dawn engines is somewhere around 850.

You can also use Eve as a gravity assist to get to Dres.
For Moho you also use Eve. Eve can easily catapult your inclination to meet that of Moho without to much hassle. Luckily there are many Moho transfer windows so that shouldn't be hard. Always try to meet Moho at it's Ap around the Sun where it's relative solar velocity is lowest so that your approach is slowest while matching it's inclination path.

[Kryxal]

Either the AN/DN are close to AP/PE, in which case you can mostly make the plane change with the ejection and circularization burns, or they're not near and mid-course corrections will be highly effective.

By the way, if going to Moho, the "easy" way is to forget about intercepting the planet to start, and just worry about reaching the orbit, with the lowest possible PE that's tangent to Moho's orbit.  Make your plane change at the same time, and set up your intercept for a later point.

[FloppyRocket]

I've been experimenting with the alex moon "Launch Window Planner" web page, along with the PreciseNode mod for getting accurate maneuvers - particularly for making fine adjustments to the time.

1) In LWP, if you pick the "ballistic" option, you get a one-burn ejection from orbit that gets you to the destination without an additional plane change burn. Your departure from one SOI and your arrival in the destination SOI will not be in either equatorial plane.

LWP gives you several critical pieces of information.

- The Year and Day and Hour of the optimum departure. Ignore the minutes and seconds.

- The ejection angle and either "to prograde" or "to retrograde".

- The burn velocity, use the "info" button to get the Prograde and Normal components

- How much total delta-v you'll need for the ejection burn and for the encounter burn to get into a circular orbit when you arrive. You can use these delta-v values for planning your rocket design.

2) To set up the ejection burn, you warp ahead in KSP to the correct Year, Day and Hour. Ignore the predicted minutes and seconds, those are just guesses by LWP because it doesn't really know what your origin orbit is exactly - just its altitude. 

3) Add a maneuver node, and key in the two components (Prograde and Normal) of the burn velocity using PreciseNode.

4) Now you use the PreciseNode time adjustments, watching the Ejection Angle until it matches the value from LWP. Note that LWP says "from prograde" and "from retrograde" instead of "to...".  Ignore the difference in terminology. You may need to adjust the time by up to a full orbit. 

At this point, the encounter markers in the map view may or may not show that you're getting an encounter.  If they don't you can typically get the markers to show an encounter if you add a second node slightly in the future. You don't need to enter anything in the node, just having a second node seems to make KSP display the encounter markers more reliably.

5) Execute the burn.

6) After the burn, add another maneuver node and use it to fine-tune the encounter distance. The farther away you are from the destination, the larger the impact of the velocity changes.

Edited April 1, 2019 by FloppyRocket

[Actually_New_KSP_Player]

1 hour ago, Kryxal said:

Either the AN/DN are close to AP/PE, in which case you can mostly make the plane change with the ejection and circularization burns, or they're not near and mid-course corrections will be highly effective.

What do you mean by "make the plane change with the ejection and circularization burns"? If you've meant doing the inclination change at the same time as burning prograde in LKO, then It will result in a big waste of fuel because If my Pe is nearby it dramastically increases the dV requirements. Maybe I could try burning at the outer node and just wait untill I get an encounter though.

 

19 minutes ago, FloppyRocket said:

I've been experimenting with the alex moon "Launch Window Planner" web page

That's all cool and stuff, but the point is that I've been using such planners when I was fairly new to the game and didn't know much about orbital mechanics and else, let alone interplanetary travel and now I just want to play the game myself. That's why I am asking.

The transfer window planner provides all the info you've mentioned, but it has great option to enter the departure date from the planner directly into the precise node and it is also in-game so you don't have to alt-tab and keep the tab open.

4 hours ago, Aeroboi said:

To Moho and Dress I always use gravity assists. 

Any guides I can read or watch about them? The only experience with gravity assists I had was getting into Minmus encounter from Mun's assist, but it's very little scale compared to interplanetary.

Been to Jool few times and his system is so populated that the gravity assists there seemed like complete mess though I heard of some magical Tylo assist I haven't looked up yet.

If I flyby a planet and burn at it's Pe prograde my final orbit's altitude decreases. Why?

How can I see my final orbit after like 2 encounters? Let's say I flyby Duna, but touching a little of Ike's SOI, after that I literally can not see what my final orbit will be.

^ Bunch of questions here.

[Aeroboi]

18 minutes ago, Actually_New_KSP_Player said:

Been to Jool few times and his system is so populated that the gravity assists there seemed like complete mess though I heard of some magical Tylo assist I haven't looked up yet.

The point in gravity assist's is not that it speeds your vessel up but changes it's trajectory relative to the greater local gravity well. Usually That is the sun, in the Joolian system it is Jool. So what you want to do is enter the target planet (on which you want to use a gravity assist "Tylo" apparently.) on the right or left side depending on whether it is left or right from jool relative to the *ecliptic (*relative to which is left and right)

So if you want to decelerate using Tylo play with the maneuver node by dragging the radial/radial out to see if the orbit line decreases after the slingshot when it's in front of Jool. You can use the maneuver node many days before entering Jool to see if Tylo will be at the correct spot to meet. If this isn't the case you should create a maneuver node so you arrive at Jool sooner or later so that you'll meet Tylo at the correct spot. In any case Tylo could be on the other side of Jool at the time of entering so you should plan ahead that way.

18 minutes ago, Actually_New_KSP_Player said:

If I flyby a planet and burn at it's Pe prograde my final orbit's altitude decreases. Why?

What do you mean?

19 minutes ago, Actually_New_KSP_Player said:

How can I see my final orbit after like 2 encounters? Let's say I flyby Duna, but touching a little of Ike's SOI, after that I literally can not see what my final orbit will be.

In the KSP settings there's a conic patch limit slider. Each number past 1 presents another calculated orbit line. So that would be Duna <> Ike but nothing else. If it were 3 you could see Dres if that were your next destination.

 

23 minutes ago, Actually_New_KSP_Player said:

Any guides I can read or watch about them? The only experience with gravity assists I had was getting into Minmus encounter from Mun's assist, but it's very little scale compared to interplanetary.

The point is to use as much fuel to make a 2/3 resonance outside of Kerbin to meet back with Kerbin to rise the Ap above the sun. You have to match orbital time period 1.5x that around the sun compared to Kerbin to meet it back two orbits around the sun. You should use Better timewarp mod. It could then slingshot you to Duna to get to Dres for very little fuel and plane change also but you'll have to calculate ahead to see if Duna is at the right place.

Here's Mark Thrimm's trip To Dress using the Kerbin <> Duna Gravity assist after refueling his SSTO at Minmus.

 

[Actually_New_KSP_Player]

4 minutes ago, Aeroboi said:

What do you mean?

I mean that if we take again Duna as an example, get an encounter with it with Pe about 55-60 kms to avoid the atmosphere, then set a node at this Pe and move it prograde we will see that our solar orbit decreases. While writing this comment I thought that relative to the Duna I've been burning prograde, but relative to the sun I've been burning somewhere near radial and that might be the case.

[Kryxal]

2 hours ago, Actually_New_KSP_Player said:

What do you mean by "make the plane change with the ejection and circularization burns"? If you've meant doing the inclination change at the same time as burning prograde in LKO, then It will result in a big waste of fuel because If my Pe is nearby it dramastically increases the dV requirements. Maybe I could try burning at the outer node and just wait untill I get an encounter though.

You'd be surprised how much delta-v you can save by combining the ejection burn with a normal/antinormal burn.  If you set up a 1000 m/s ejection burn, then add maybe 750 m/s normal, you'll end up with a 1250 m/s burn.  Also, you'll probably have to pull the prograde component back some, this sort of thing tends to mean you get a higher AP or lower PE.

Alternatively, if you just want to cause an intercept, a burn at about 90 degrees out is perfect for fine-tuning that.  It may not match the plane as well, but it's good for getting you there.

[Zhetaan]

16 hours ago, Actually_New_KSP_Player said:

What do you mean by "make the plane change with the ejection and circularization burns"? If you've meant doing the inclination change at the same time as burning prograde in LKO, then It will result in a big waste of fuel because If my Pe is nearby it dramastically increases the dV requirements. Maybe I could try burning at the outer node and just wait untill I get an encounter though.

 

14 hours ago, Kryxal said:

You'd be surprised how much delta-v you can save by combining the ejection burn with a normal/antinormal burn.

@Kryxal is correct here, for certain orbits in certain circumstances, but not all.  One thing to remember about plane change burns (and radial burns, too) is that they don't change the energy of the orbit; they change the direction.  The calculations for them are different from what you would otherwise use.  To wit:

To go from one orbit altitude to another in a Hohmann transfer requires solving a variation of the vis-viva equation, which tells orbital velocity at a particular altitude and for an orbit of a given semi-major axis.  In other words, you solve the vis-viva equation for a circular orbit at one altitude, solve it for the Hohmann transfer orbit where its apsis is at that altitude, and the difference in velocities is the delta-V needed to go from one orbit to the other.  Do the same thing when you reach the other end of the transfer, and the total for both burns is the total cost.  However, the usual rules for efficient orbit modifications do not apply to normal burns (or radial burns, for that matter).  The details of calculating it are in the spoiler.

Spoiler

I won't go through an example calculation; that's not the point of my response.  I will, however, show you the equation:

v² = μ [(2 / r) - (1 / a)]

Where:

v = orbital velocity
μ = standard gravitational parameter (a constant for a given sphere of influence; for Kerbin, it's 3.5316 x 10¹² m³/s²)
r = current orbital radius
a = semi-major axis of the orbit

Note that nothing in this equation cares about your inclination.  That means two useful things:  first, the inclination is irrelevant to orbital speed, so an object in circular orbit of 75 km about Kerbin is going to be moving at 2287 m/s whether that orbit is equatorial or polar.  Second, it means that the vis-viva equation is useless to tell you the delta-V to go from the equatorial to the polar orbit, because the altitude and semi-major axis don't change.  The vis-viva equation says that they're the same orbit, so the cost is zero, but obviously that's not true.

The key to solving this is to understand vector mathematics and trigonometry.  Specifically, you are travelling at a given speed in a given direction; that's what velocity is.  If you want to change the direction but not the speed (necessary if you want to reach the point of intersection with something else's orbit without changing the altitude), then you are asking to rotate the velocity to a new direction without changing its magnitude.  Imagine, for example, that your vessel is on an orbit, and we're looking at it so that its current direction is straight up along the y-axis.  The vessel on that orbit follows a curved path, but its velocity (specifically, its tangential velocity) is always in a straight line.  That line corresponds to the prograde direction.  This is all stuff that you probably know, of course, but I think it important to revisit it from a vector maths point-of-view.

I drew a picture to assist, so please kindly fear my artistic skill:

The circle is only present to prove that the initial velocity and final velocity, the bright red and green arrows, respectively, are of the same magnitude.  Because both vectors start at the origin and end on a circle also centred on the origin, the vectors are both radii and thus must be the same length, i.e., magnitude.

To change the direction but not the magnitude of the initial velocity is as simple (on paper, at least) as rotating it through some angle.  If you want to incline your orbit by five degrees, then the angle would be five degrees, and so forth.  Since your initial velocity is known (the vis-viva equation will give you a reliable answer for that) and your final velocity is the same magnitude, the burn is the difference in direction.  Going back to the image, a change of five degrees (let's assume that the x-axis is the normal direction, so, clockwise) will point the new vector a bit to the right.  The burn necessary to make this change is the same as the vector connecting the endpoints, going from the end of the original vector to the end of the new one.  This has to do with vector addition, but the important part is that many vectors can be added together by starting the next vector at the endpoint of the previous vector, and wherever that combination ends, you can draw one vector that goes from the origin to that ultimate endpoint (the green arrow, in this case), and that single vector is the same as the sum of all the components.

If the vessel were starting from the ground, it wound make sense to launch directly to the green vector, but since we assume it is already in orbit (and thus already have the red vector velocity), the vessel needs only to burn the same magnitude and direction as the dark red vector to effect the change to the green one.  Solving for this burn is the same as solving for the third side of a triangle where you know the angle between the two known sides, which means that we can use the Law of Cosines to get the burn magnitude.

Since a 75 km orbit of Kerbin has an orbital velocity of 2,287 m/s, that is our side length for the bright red and green vectors.  The angle is the five-degree inclination change.  The Law of Cosines is as follows:

c² = a² + b² - 2ab * cos C

Where:
c = dark red vector magnitude
a = bright red vector magnitude
b = green vector magnitude
C = desired inclination

Since a and b are the same magnitude in this case, we can equate them:

c² = 2a² - 2a² * cos C
c² = 2a² * (1 - cos C)
c = a √[2 * (1 - cos C)]

We can actually simplify this further with the use of a half-angle formula:

c = 2a √([2 * (1 - cos C)] / 4)
c = 2a √[(1 - cos C) / 2]
c = 2a * sin (C / 2)

But that is mostly academic.  The important part is that the calculation can be modified to make sense for space travel in the following way:

dV = 2v * sin (θ / 2)
or:
dV = v √[2 * (1 - cos θ)]

Where dV is the delta-V needed for the burn, v is the orbital velocity, and θ is the desired inclination change.  The other necessary part is to know the direction of the burn, but that is easily solved:  identical initial and final velocities make an isosceles triangle, and triangle interior angles always add to 180°, so subtract the inclination change from 180° and divide the difference by two.  For this example, the resultant angle is half of 175°, or 87.5°, but remember that that is the angle as measured from the downwards or retrograde direction.

The relationship between the velocity and the cost to change the direction of that velocity is linear:  in other words, to get the most out of the change, it is better to do so when the velocity is lowest--this much, you obviously know.  However, you should also take into account the larger picture of your solar orbit.  Periapsis is normally the absolute worst place to change inclination, but even though the velocity there is high, have you considered what planetary orbital velocity is?  Consider a transfer to Eve:  Eve's orbit is inclined by 2.1°, but Eve's average orbital velocity is 10,920 m/s.  Kerbin's is 9,284.5 m/s.  Add to that that Kerbin is at the apoapsis of the transfer to Eve, and it begins to make sense to change inclination there, local periapsis or not.

If, on the other hand, you are transferring to Dres, then things are a bit different.  Dres is inclined by 5° but it has an average orbital velocity of 5,415 m/s.  Since Kerbin sits at the periapsis of the transfer orbit and the inclination is large, a mid-course correction partway to Dres makes more sense, in the amount of about 500 m/s.

On the gripping hand, it sometimes doesn't make sense to change inclination at all.  Moho is inclined by 7° and has an orbital velocity that varies from over 12 km/s to over 18 km/s.  In that case, it can be cheaper to forego the launch window and instead launch to encounter Moho's ascending node, follow with a resonance burn once there to encounter Moho itself when it next comes round to that node, and complete the transfer by circularisation at the encounter as normal.

Edited April 2, 2019 by Zhetaan