A key part of an efficient transfer is launching at the right time. There are lists for these in some cases, but so far they’ve been rather incomplete. This aims to fix it.

If you’re unsure on what a phase angle is, it’s the angle between the body you’re launching from and the one you’re launching towards, as seen from whatever both are orbiting around. I’m using + when the target is ahead of you, and - when it’s behind for reasons that will be apparent shortly.

Derivation

These are relatively simple to calculate. From Kepler’s 3rd law, \(T^2 ~ a^3\)

So the period of a transfer orbit is \(T_x ~ a^{3/2} = (\frac{a_1 + a_2}{2})^{3/2}\)

And the distance covered by the target orbit during 1/2 of the transfer orbit period (in fraction of the outer orbit’s circle) is \(\frac{1/T_2}{2/T_x} = \frac{T_x /2}{T_2} = 1/2 (\frac{a_1/a_2 + 1}{2})^{3/2}\)

The transfer orbit will cover 180° or π radians in that time, so we want a launch angle that’s said 180° minus the distance covered by the target planet. Or: \(θ = π(1-(\frac{a_1/a_2-1}{2})^{3/2}) = 180°(1-(\frac{a_1/a_2-1}{2})^{3/2})\)

Tables

Some values are (as you can see) a bit silly, so ones that are modulo 360° are also provided. Even so, many inwards transfers don’t really care about timing, as you spend so much in flight compared with the target that you can just tweak your orbit slightly to arrive. Outwards transfers eventually converge on \(π(1-\frac{1}{2\sqrt{2}}) ~= 2.03\) radians, or ~116° for sufficiently distant orbits (though you rarely reach that).

The raw data / calculations are here.

Kerbal Space Program (KSP) Planets

Note that Moho, Dres, and Eeloo all have meaningfully large inclinations and eccentricities, so actual transfers may vary significantly.

From\To Moho Eve Kerbin Duna Dres Jool Eeloo
Moho   58.94 76.05 90.64 103.7 108.9 110.7
Eve -129.1   36.07 66.07 92.04 102.2 105.7
Kerbin -251.8 -54.13   44.36 82.06 96.58 101.4
Duna -518.3 -168.7 -75.19   62.21 85.52 93.19
Dres -1469.9 -564.5 -329.7 -145.8   51.95 68.52
Jool -3177.6 -1258.5 -768.7 -391.1 -99.83   31.01
Eeloo -4729.8 -1882.5 -1160.3 -607.1 -185.4 -43.49  

Cleaned up a bit, with all angles below 180°:

From\To Moho Eve Kerbin Duna Dres Jool Eeloo
Moho   58.94 76.05 90.64 103.7 108.9 110.7
Eve -129.1   36.07 66.07 92.04 102.2 105.7
Kerbin 108.2 -54.13   44.36 82.06 96.58 101.4
Duna -158.3 -168.7 -75.19   62.21 85.52 93.19
Dres -29.86† 155.5 30.32 -145.8   51.95 68.52
Jool 62.38† -178.5† -48.65† -31.06 -99.83   31.01
Eeloo -49.75† -82.54† -80.33† 112.9 174.6 -43.49  

†Sufficently large (>720°) that I’m assuming that there’s a lot of leeway

KSP Moons

Minmus has notable inclination, Bop and Pol have notable inclination and eccentricity.

From\To Mun Minmus
Mun   90.49
Minmus -513.8  
From\To Laythe Vall Tylo Bop Pol
Laythe   47.57 74.94 95.13 101.4
Vall -84.86   47.57 81.75 92.14
Tylo -240.3 -84.87   59.20 76.74
Bop -692.2 -324.9 -130.4   37.15
Pol -1158.0 -567.8 -259.4 -56.60  

And with all angles within ±180°:

From\To Mun Minmus
Mun   90.49
Minmus -153.8  
From\To Laythe Vall Tylo Bop Pol
Laythe   47.57 74.94 95.13 101.4
Vall -84.86   47.57 81.75 92.14
Tylo 119.7 -84.87   59.20 76.74
Bop 27.79 35.11 -130.38   37.15
Pol -77.97† 152.16 100.6 -56.60  

Juno: New Origins (JNO) Planets

Handrew’s Comet has a great deal of eccentricity (as does Cladh, though it’s so far out that you shouldn’t be using Hohmann-type transfers if you want reasonable travel times.)

From\To Vulco Sergeaa Droo Cylero Handrew’s Comet Tydos Urados Cladh
Vulco   55.14 82.19 95.40 108.3 110.2 114.7 116.3
Sergeaa -112.3   53.00 78.11 101.9 105.4 113.4 116.2
Droo -331.7 -103.8   47.57 91.06 97.31 111.3 116.1
Cylero -705.2 -275.4 -84.86   74.94 85.36 108.3 116.0
Handrew’s Comet -2821.5 -1216.2 -531.5 -240.3   31.01 95.49 115.50
Tydos -4200.3 -1820.1 -812.3 -387.8 -43.49   88.62 115.2
Urados -28598.3 -12352.7 -5605.6 -2839.8 -710.0 -461.5   112.2
Cladh -3130545.0 -1338297.6 -599627.3 -300316.0 -75553.3 -50518.3 -7379.7  

With more tractable angles:

From\To Vulco Sergeaa Droo Cylero Handrew’s Comet Tydos Urados Cladh
Vulco   55.14 82.19 95.40 108.3 110.2 114.7 116.3
Sergeaa -112.3   53.00 78.11 101.9 105.4 113.4 116.2
Droo 28.30 -103.8   47.57 91.06 97.31 111.3 116.1
Cylero 14.82 84.64 -84.86   74.94 85.36 108.3 116.0
Handrew’s Comet 58.45† -136.2† -171.5 119.7   31.01 95.49 115.50
Tydos 119.7† 20.09† -92.34† -27.83 -43.49   88.62 115.2
Urados -158.3† -112.7† 154.4† 40.23† 10.00 101.5   112.2
Cladh 14.98† -177.6† 132.7† -76.04† 46.67† -118.3† -179.7†  

JNO Moons

TT is in a retrograde orbit (inclination 2.8 radians or ~160°), so is treated as 180° for this (flipping a sign so that the angle is 180° plus the distance the target planet travcels instead of minus). In reality its inclination is still large enough that this doesn’t work well.

Herma has a large inclination (0.8 radians, or ~46°) and eccentricity (0.2), so transfers between it and Niobe won’t really follow these.

While they’re at an angle with respect to the rest of the system, all of Urados’ moons are pretty much coplanar.

From\To Brigo Luna TT
Brigo   47.57 282.0
Luna -84.86   307.2
TT 634.1 463.1  
From\To Niobe Herma
Niobe   67.03
Herma -175.0  
From\To Nebra Miros Orcus
Nebra   66.51 84.63
Miros -171.6   44.17
Orcus -373.8 -74.65  
From\To Jastrus Boreas Taurus Hypatchion
Jastrus   105.7 112.8 113.7
Boreas -1893.1   81.48 90.28
Taurus -9350.2 -320.8   31.33
Hypatchion -14003.7 -507.5 -44.12  

And finally with smaller angles (constrained to ±180°):

From\To Brigo Luna TT
Brigo   47.57 -78.01
Luna -84.86   -52.82
TT -85.91 103.1  
From\To Niobe Herma
Niobe   67.03
Herma -175.0  
From\To Nebra Miros Orcus
Nebra   66.51 84.63
Miros -171.6   44.17
Orcus -13.84 -74.65  
From\To Jastrus Boreas Taurus Hypatchion
Jastrus   105.7 112.8 113.7
Boreas -93.14†   81.48 90.28
Taurus 9.82† 39.16   31.33
Hypatchion 36.26† -147.5 -44.12  

Why these tables?

It seemed to be a gap in information. KSP has a web app for this (and a porkchop plot calculator), but having the tables in one place is convenient. Also, there appears to be no equivalent in JNO.