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.
