# How can you calculate the true anomalies for which a transfer between two coaxial orbits are possible (with only a single impulse)?

I was working on the mathematics for transfers between coplanar/coaxial elliptical orbits, and I noticed that for a single instant-impulse maneuver there are some true anomalies at which you cannot get to the desired orbit. For instance, for the orbits defined below:

Start: 20,000 apogee by 10,000 perigee

End: 20,000 apogee by 6738 (surface) perigee

If you with to get the most bang for your buck, a Hohmann transfer can be done at apogee for 0.5405 km/s delta-v. Below is a list for the delta-v requirement for the transfer between the start and end orbits by the current true anomaly of the start orbit (in degrees). In addition to this I've added the required velocity vectors for the impulse at the true anomaly on the start orbit:

0   NaN
...
55  NaN
56  52.308374094479305  (-0.90042,57.06208,0)[57.06918292828537]
57  31.803682879691532  (-1.40858,36.44501,0)[36.472225212004105]
58  24.335719991149062  (-1.77431,28.88671,0)[28.941148785963147]
59  20.185346289370077  (-2.0741,24.65412,0)[24.74121364580215]
60  17.458629196557723  (-2.33327,21.84897,0)[21.97319776526239]
61  15.49463027623545   (-2.56395,19.80832,0)[19.973568881729555]
62  13.994622096868948  (-2.77306,18.23241,0)[18.44208612612248]
63  12.801353915714838  (-2.96501,16.96334,0)[17.220518817954197]
64  11.823173132650561  (-3.14278,15.90908,0)[16.21653403962397]
65  11.002598758671676  (-3.30849,15.0119,0)[15.372154310767074]
66  10.301526580052014  (-3.46374,14.23351,0)[14.648899266887089]
67  9.693575464692366   (-3.60975,13.54743,0)[14.020094423402247]
68  9.159829672837434   (-3.74747,12.93466,0)[13.46658994610399]
69  8.686329349030009   (-3.87767,12.38121,0)[12.974235675171858]
70  8.26251990506506    (-4.00098,11.8765,0)[12.532319947400646]
71  7.880255481655833   (-4.11792,11.41236,0)[12.132565613360503]
72  7.533136832512499   (-4.2289,10.9824,0)[11.768462529638915]
73  7.216058666467891   (-4.33431,10.58152,0)[11.434810904037821]
74  6.924892426994145   (-4.43445,10.20562,0)[11.127401051904656]
75  6.65625910958260    (-4.52959,9.85133,0)[10.8427838861782]
76  6.407363412101789   (-4.61998,9.51589,0)[10.578103248032708]
77  6.17587057617189    (-4.7058,9.19698,0)[10.330971305674131]
78  5.959813521495532   (-4.78724,8.89268,0)[10.099374530895519]
79  5.757521850516562   (-4.86447,8.60134,0)[9.881601764213485]
80  5.567566891022103   (-4.93761,8.32157,0)[9.676188487433608]
81  5.388718667499432   (-5.00681,8.05219,0)[9.481873158179319]
82  5.219911860251407   (-5.07217,7.79216,0)[9.297562638032328]
83  5.060218616930607   (-5.1338,7.54059,0)[9.122304558057255]
84  4.908826645565717   (-5.19179,7.29672,0)[8.955265034652426]
85  4.765021419299695   (-5.24622,7.05987,0)[8.795710553355882]
86  4.628171611975426   (-5.29718,6.82944,0)[8.642993129819976]
87  4.497717094357069   (-5.34473,6.60492,0)[8.496538069852207]
88  4.373158976121882   (-5.38895,6.38586,0)[8.355833807330047]
89  4.254051294537515   (-5.42988,6.17185,0)[8.220423415791497]
90  4.139994037883991   (-5.4676,5.96253,0)[8.089897477598656]
91  4.030627257879442   (-5.50216,5.75758,0)[7.963888061522148]
92  3.9256260760925885  (-5.5336,5.55673,0)[7.842063610916328]
93  3.824696428507533   (-5.56197,5.35972,0)[7.72412458431665]
94  3.727571422904664   (-5.58731,5.16633,0)[7.609799721175589]
95  3.6340082076319593  (-5.60968,4.97636,0)[7.498842829677989]
96  3.543785269213139   (-5.62911,4.78962,0)[7.391030012705348]
97  3.4567000912311014  (-5.64563,4.60597,0)[7.286157263221345]
98  3.372567118906868   (-5.65929,4.42526,0)[7.184038372506629]
99  3.2912159834254155  (-5.67012,4.24735,0)[7.084503104446573]
100 3.2124899478441695  (-5.67815,4.07214,0)[6.987395596980388]
101 3.1362445427440795  (-5.68342,3.89952,0)[6.89257295824501]
102 3.062346364946846   (-5.68597,3.72941,0)[6.799904030195514]
103 2.990672016857931   (-5.68581,3.56172,0)[6.709268296791361]
104 2.921107167485339   (-5.68299,3.39637,0)[6.620554917388462]
105 2.853545719073352   (-5.67753,3.23332,0)[6.5336618689177675]
106 2.787889065690697   (-5.66947,3.07249,0)[6.448495182874643]
107 2.7240454321161596  (-5.65883,2.91383,0)[6.36496826518423]
108 2.6619292830417103  (-5.64565,2.75731,0)[6.283001288716415]
109 2.6014607940236756  (-5.62994,2.60289,0)[6.2025206496614675]
110 2.5425653768018126  (-5.61175,2.45052,0)[6.1234584801895995]
111 2.4851732526130563  (-5.59109,2.30018,0)[6.045752210843889]
112 2.429219067981678   (-5.568,2.15185,0)[5.969344176987303]
113 2.374641548196164   (-5.5425,2.0055,0)[5.894181264366682]
114 2.321383184305478   (-5.51463,1.86112,0)[5.820214589490067]
115 2.2693899500013632  (-5.48441,1.71868,0)[5.7473992110566865]
116 2.218611045211835   (-5.45187,1.57818,0)[5.675693869144521]
117 2.168998663627105   (-5.41703,1.43961,0)[5.605060749261414]
118 2.120507781721143   (-5.37993,1.30296,0)[5.535465268711247]
119 2.0730959671292797  (-5.3406,1.16823,0)[5.4668758830255095]
120 2.0267232045004917  (-5.29905,1.03541,0)[5.3992639104689895]
121 1.981351737168596   (-5.25533,0.9045,0)[5.332603372852457]
122 1.9369459231839612  (-5.20947,0.7755,0)[5.266870851079487]
123 1.8934721044211447  (-5.16148,0.64841,0)[5.202045354023695]
124 1.8508984876303465  (-5.1114,0.52324,0)[5.138108199478922]
125 1.8091950364364846  (-5.05926,0.4,0)[5.0750429060526345]
126 1.7683333734095759  (-5.00508,0.27868,0)[5.012835094983139]
127 1.728286691437784   (-4.94891,0.15929,0)[4.951472400956974]
128 1.6890296737306891  (-4.89077,0.04185,0)[4.890944391085781]
129 1.6505384218669321  (-4.83068,-0.07365,0)[4.831242491272758]
130 1.6127903913797828  (-4.76869,-0.18718,0)[4.772359919260453]
131 1.5757643344450094  (-4.70482,-0.29875,0)[4.714291623702768]
132 1.539440249302408   (-4.6391,-0.40834,0)[4.657034228648327]
133 1.5037993361025341  (-4.57156,-0.51593,0)[4.6005859828582]
134 1.4688239589274672  (-4.50225,-0.62152,0)[4.544946713410777]
135 1.4344976137873722  (-4.43118,-0.7251,0)[4.490117783069955]
136 1.4008049024450548  (-4.3584,-0.82665,0)[4.436102050910883]
137 1.3677315119681877  (-4.28393,-0.92617,0)[4.382903835710386]
138 1.3352641999545698  (-4.20781,-1.02363,0)[4.330528881617806]
139 1.3033907854188609  (-4.13007,-1.11902,0)[4.278984325626156]
140 1.2721001453707586  (-4.05075,-1.21234,0)[4.228278666364501]
141 1.2413822171539115  (-3.96988,-1.30357,0)[4.178421733729903]
142 1.2112280066514045  (-3.88749,-1.39269,0)[4.129424658872381]
143 1.1816296024976052  (-3.80362,-1.47969,0)[4.0812998440391635]
144 1.1525801964658349  (-3.71831,-1.56456,0)[4.034060931775983]
145 1.1240741102258487  (-3.63158,-1.64729,0)[3.987722772973585]
146 1.0961068286824012  (-3.54348,-1.72786,0)[3.9423013932382727]
147 1.0686750401142948  (-3.45404,-1.80626,0)[3.8978139570564205]
148 1.041776683328122   (-3.3633,-1.88247,0)[3.8542787292159004]
149 1.0154110020192884  (-3.27129,-1.95649,0)[3.811715032943064]
150 0.9895786064888148  (-3.17805,-2.02829,0)[3.7701432042134884]
151 0.9642815427912089  (-3.08362,-2.09788,0)[3.7295845416995106]
152 0.939523369278632   (-2.98803,-2.16523,0)[3.6900612518287246]
153 0.9153092403484342  (-2.89133,-2.23033,0)[3.6515963884468783]
154 0.8916459969839219  (-2.79354,-2.29318,0)[3.614213786606987]
155 0.8685422633862228  (-2.69471,-2.35376,0)[3.5779379900458017]
156 0.8460085486133817  (-2.59488,-2.41205,0)[3.5427941719602107]
157 0.8240573516517983  (-2.49408,-2.46805,0)[3.508808048760981]
158 0.8027032677271861  (-2.39235,-2.52176,0)[3.4760057865606986]
159 0.781963092898264   (-2.28974,-2.57315,0)[3.4444139002473975]
160 0.7618559230521552  (-2.18627,-2.62221,0)[3.414059145105819]
161 0.7424032423273372  (-2.082,-2.66895,0)[3.384968401074392]
162 0.7236289947334964  (-1.97695,-2.71334,0)[3.357168549867104]
163 0.7055596313407951  (-1.87117,-2.75539,0)[3.330686345344475]
164 0.6882241239282614  (-1.7647,-2.79508,0)[3.3055482776844776]
165 0.6716539345042171  (-1.65758,-2.8324,0)[3.2817804320799624]
166 0.6558829287850911  (-1.54985,-2.86735,0)[3.259408342869978]
167 0.6409472207457108  (-1.44154,-2.89992,0)[3.238456844194252]
168 0.6268849350030334  (-1.33271,-2.93011,0)[3.218949918437519]
169 0.6137358743911174  (-1.22338,-2.9579,0)[3.2009105438976357]
170 0.6015410819896593  (-1.1136,-2.9833,0)[3.184360543262455]
171 0.5903422904305861  (-1.00341,-3.00629,0)[3.169320434608427]
172 0.5801812567952165  (-0.89285,-3.02687,0)[3.155809286732937]
173 0.5710989889150466  (-0.78196,-3.04504,0)[3.1438445806959803]
174 0.5631348782022509  (-0.67079,-3.0608,0)[3.1334420794702758]
175 0.5563257646767508  (-0.55937,-3.07414,0)[3.124615707577699]
176 0.5507049706063348  (-0.44774,-3.08506,0)[3.1173774425215455]
177 0.5463013487289191  (-0.33595,-3.09355,0)[3.111737219707578]
178 0.5431383977446799  (-0.22404,-3.09962,0)[3.107702852382802]
179 0.541233500074307   (-0.11204,-3.10326,0)[3.1052799679124177]
180 0.5405973336216916  (0,-3.10447,0)[3.10447196146722]
181 0.541233500074307   (0.11204,-3.10326,0)[3.1052799679124177]
182 0.5431383977446765  (0.22404,-3.09962,0)[3.107702852382802]
183 0.5463013487289223  (0.33595,-3.09355,0)[3.111737219707578]
184 0.5507049706063348  (0.44774,-3.08506,0)[3.1173774425215455]
185 0.5563257646767508  (0.55937,-3.07414,0)[3.124615707577699]
186 0.5631348782022509  (0.67079,-3.0608,0)[3.133442079470277]
187 0.5710989889150435  (0.78196,-3.04504,0)[3.1438445806959807]
188 0.5801812567952196  (0.89285,-3.02687,0)[3.155809286732937]
189 0.5903422904305861  (1.00341,-3.00629,0)[3.169320434608427]
190 0.6015410819896593  (1.1136,-2.9833,0)[3.1843605432624535]
191 0.6137358743911203  (1.22338,-2.9579,0)[3.2009105438976357]
192 0.6268849350030306  (1.33271,-2.93011,0)[3.218949918437518]
193 0.6409472207457108  (1.44154,-2.89992,0)[3.238456844194251]
194 0.6558829287850911  (1.54985,-2.86735,0)[3.259408342869978]
195 0.6716539345042171  (1.65758,-2.8324,0)[3.2817804320799624]
196 0.6882241239282588  (1.7647,-2.79508,0)[3.305548277684479]
197 0.7055596313407977  (1.87117,-2.75539,0)[3.330686345344473]
198 0.723628994733494   (1.97695,-2.71334,0)[3.357168549867104]
199 0.7424032423273372  (2.082,-2.66895,0)[3.3849684010743926]
200 0.7618559230521575  (2.18627,-2.62221,0)[3.4140591451058193]
201 0.7819630928982686  (2.28974,-2.57315,0)[3.4444139002473966]
202 0.8027032677271839  (2.39235,-2.52176,0)[3.476005786560699]
203 0.824057351651796   (2.49408,-2.46805,0)[3.5088080487609803]
204 0.8460085486133796  (2.59488,-2.41205,0)[3.5427941719602094]
205 0.8685422633862268  (2.69471,-2.35376,0)[3.5779379900458017]
206 0.89164599698392    (2.79354,-2.29318,0)[3.614213786606988]
207 0.9153092403484303  (2.89133,-2.23033,0)[3.6515963884468765]
208 0.9395233692786245  (2.98803,-2.16523,0)[3.6900612518287237]
209 0.9642815427912089  (3.08362,-2.09788,0)[3.7295845416995084]
210 0.9895786064888148  (3.17805,-2.02829,0)[3.770143204213488]
211 1.0154110020192866  (3.27129,-1.95649,0)[3.8117150329430634]
212 1.0417766833281201  (3.3633,-1.88247,0)[3.854278729215901]
213 1.0686750401142964  (3.45404,-1.80626,0)[3.8978139570564214]
214 1.0961068286824027  (3.54348,-1.72786,0)[3.9423013932382727]
215 1.1240741102258502  (3.63158,-1.64729,0)[3.987722772973584]
216 1.1525801964658318  (3.71831,-1.56456,0)[4.034060931775983]
217 1.1816296024976052  (3.80362,-1.47969,0)[4.081299844039164]
218 1.2112280066514016  (3.88749,-1.39269,0)[4.129424658872378]
219 1.241382217153917   (3.96988,-1.30357,0)[4.178421733729905]
220 1.2721001453707586  (4.05075,-1.21234,0)[4.228278666364499]
221 1.3033907854188582  (4.13007,-1.11902,0)[4.278984325626153]
222 1.3352641999545698  (4.20781,-1.02363,0)[4.330528881617803]
223 1.3677315119681797  (4.28393,-0.92617,0)[4.382903835710383]
224 1.4008049024450548  (4.3584,-0.82665,0)[4.436102050910883]
225 1.4344976137873822  (4.43118,-0.7251,0)[4.490117783069959]
226 1.4688239589274672  (4.50225,-0.62152,0)[4.544946713410777]
227 1.5037993361025317  (4.57156,-0.51593,0)[4.600585982858198]
228 1.5394402493024104  (4.6391,-0.40834,0)[4.657034228648329]
229 1.5757643344450116  (4.70482,-0.29875,0)[4.714291623702768]
230 1.6127903913797805  (4.76869,-0.18718,0)[4.77235991926045]
231 1.6505384218669277  (4.83068,-0.07365,0)[4.8312424912727545]
232 1.689029673730683   (4.89077,0.04185,0)[4.890944391085779]
233 1.7282866914377901  (4.94891,0.15929,0)[4.9514724009569795]
234 1.7683333734095779  (5.00508,0.27868,0)[5.012835094983138]
235 1.8091950364364846  (5.05926,0.4,0)[5.075042906052634]
236 1.8508984876303523  (5.1114,0.52324,0)[5.138108199478925]
237 1.8934721044211429  (5.16148,0.64841,0)[5.202045354023695]
238 1.9369459231839592  (5.20947,0.7755,0)[5.266870851079488]
239 1.981351737168589   (5.25533,0.9045,0)[5.332603372852452]
240 2.0267232045004953  (5.29905,1.03541,0)[5.399263910468993]
241 2.073095967129275   (5.3406,1.16823,0)[5.466875883025505]
242 2.120507781721143   (5.37993,1.30296,0)[5.535465268711248]
243 2.1689986636271033  (5.41703,1.43961,0)[5.605060749261413]
244 2.218611045211835   (5.45187,1.57818,0)[5.6756938691445225]
245 2.2693899500013632  (5.48441,1.71868,0)[5.747399211056685]
246 2.3213831843054824  (5.51463,1.86112,0)[5.820214589490069]
247 2.374641548196161   (5.5425,2.0055,0)[5.89418126436668]
248 2.4292190679816823  (5.568,2.15185,0)[5.969344176987309]
249 2.485173252613055   (5.59109,2.30018,0)[6.045752210843889]
250 2.542565376801814   (5.61175,2.45052,0)[6.1234584801895995]
251 2.6014607940236756  (5.62994,2.60289,0)[6.2025206496614675]
252 2.661929283041709   (5.64565,2.75731,0)[6.283001288716414]
253 2.7240454321161622  (5.65883,2.91383,0)[6.36496826518423]
254 2.7878890656906994  (5.66947,3.07249,0)[6.448495182874644]
255 2.853545719073351   (5.67753,3.23332,0)[6.533661868917765]
256 2.921107167485337   (5.68299,3.39637,0)[6.620554917388462]
257 2.9906720168579284  (5.68581,3.56172,0)[6.7092682967913575]
258 3.062346364946846   (5.68597,3.72941,0)[6.799904030195518]
259 3.1362445427440795  (5.68342,3.89952,0)[6.89257295824501]
260 3.2124899478441695  (5.67815,4.07214,0)[6.987395596980388]
261 3.291215983425405   (5.67012,4.24735,0)[7.084503104446565]
262 3.372567118906855   (5.65929,4.42526,0)[7.184038372506619]
263 3.456700091231106   (5.64563,4.60597,0)[7.28615726322135]
264 3.543785269213143   (5.62911,4.78962,0)[7.391030012705352]
265 3.6340082076319558  (5.60968,4.97636,0)[7.4988428296779865]
266 3.7275714229046524  (5.58731,5.16633,0)[7.609799721175582]
267 3.82469642850752    (5.56197,5.35972,0)[7.724124584316638]
268 3.9256260760925885  (5.5336,5.55673,0)[7.842063610916328]
269 4.030627257879442   (5.50216,5.75758,0)[7.963888061522148]
270 4.139994037883989   (5.4676,5.96253,0)[8.089897477598656]
271 4.254051294537511   (5.42988,6.17185,0)[8.220423415791496]
272 4.373158976121888   (5.38895,6.38586,0)[8.35583380733005]
273 4.497717094357069   (5.34473,6.60492,0)[8.496538069852207]
274 4.628171611975432   (5.29718,6.82944,0)[8.642993129819981]
275 4.765021419299682   (5.24622,7.05987,0)[8.795710553355867]
276 4.90882664556572    (5.19179,7.29672,0)[8.95526503465243]
277 5.060218616930593   (5.1338,7.54059,0)[9.122304558057245]
278 5.219911860251407   (5.07217,7.79216,0)[9.297562638032328]
279 5.388718667499431   (5.00681,8.05219,0)[9.481873158179319]
280 5.567566891022096   (4.93761,8.32157,0)[9.676188487433603]
281 5.757521850516546   (4.86447,8.60134,0)[9.881601764213471]
282 5.959813521495539   (4.78724,8.89268,0)[10.099374530895524]
283 6.17587057617189    (4.7058,9.19698,0)[10.330971305674131]
284 6.407363412101794   (4.61998,9.51589,0)[10.578103248032711]
285 6.656259109582563   (4.52959,9.85133,0)[10.842783886178164]
286 6.924892426994136   (4.43445,10.20562,0)[11.127401051904647]
287 7.216058666467901   (4.33431,10.58152,0)[11.434810904037832]
288 7.533136832512492   (4.2289,10.9824,0)[11.768462529638908]
289 7.880255481655807   (4.11792,11.41236,0)[12.132565613360482]
290 8.262519905065092   (4.00098,11.8765,0)[12.532319947400676]
291 8.68632934903002    (3.87767,12.38121,0)[12.974235675171869]
292 9.159829672837434   (3.74747,12.93466,0)[13.46658994610399]
293 9.693575464692358   (3.60975,13.54743,0)[14.02009442340224]
294 10.301526580051997  (3.46374,14.23351,0)[14.648899266887074]
295 11.002598758671708  (3.30849,15.0119,0)[15.372154310767106]
296 11.823173132650535  (3.14278,15.90908,0)[16.21653403962394]
297 12.801353915714968  (2.96501,16.96334,0)[17.22051881795433]
298 13.99462209686898   (2.77306,18.23241,0)[18.442086126122515]
299 15.494630276235384  (2.56395,19.80832,0)[19.973568881729488]
300 17.458629196557723  (2.33327,21.84897,0)[21.97319776526239]
301 20.185346289369996  (2.0741,24.65412,0)[24.741213645802066]
302 24.335719991148668  (1.77431,28.88671,0)[28.941148785962753]
303 31.803682879691042  (1.40858,36.44501,0)[36.472225212003615]
304 52.30837409448151   (0.90042,57.06208,0)[57.06918292828758]
305 NaN
...
360 NaN


And you can see the animation of the transfer orbit changing here, the further from apogee we go for this impulse, the more expensive, obviously. For this specific orbital transfer it seems that the transfer window is from 56 degrees to 304 degrees, give or take.

If you watch the animation, near the end there's a cut-off which seems to coincide with something. I'm guessing the transfer is impossible if you exceed a certain parameter, but I do not know what. I'm also assuming that cut-off is before 52km/s deltaV required for a transfer at 54 degrees (because that's near 4.5x the escape velocity of earth).

My question is:

How can I mathematically know at what the valid range of true anomalies for a given transfer is without having to calculate an approximation through trial and error? I would prefer you simplify orbits to a 2D model with coplanar/coaxial orbits.

• @uhoh All coaxial means is orbits that have their semimajor axis aligned. – Magic Octopus Urn Oct 12 '18 at 8:28
• can you double check the edited first sentence, does this look okay? – uhoh Oct 12 '18 at 9:04
• I still don't understand the question. The two orbits have apogee at 20,000 km. This is the only point where these coplanar/coaxial orbits ever intersect. There is only one possible "single instant-impulse maneuver" and that's at their mutual apogee point. – uhoh Oct 12 '18 at 9:42
• I think the asker is limiting the choices to only making a burn along the direction the spacecraft is currently moving, and ignoring the second burn required to match the orbits at the target, if that's the case, then the reachable positions on the target orbit can be found by drawing a line tangent to the sourceorbit at the point where the impulse will be made, and all points on the target orbit that are on the same side of the tangent line as the body being orbited can be intercepted in that fashion. – notovny Oct 12 '18 at 9:49
• @uhoh I sort of asked this at a bad time, I'm in Virginia visiting some relatives, I'll try to post my calculations and some updates when I get back. What he said gave me an idea but I don't have my workhorse PC with me – Magic Octopus Urn Oct 13 '18 at 13:53