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A numerical mapping of energy gains in a powered Swing-By maneuver

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Abstract

The present paper studies the effects of a powered Swing-By maneuver, considering the particular and important situations where there are energy gains for the spacecraft. The objective is to map the energy variations obtained from this maneuver as a function of the three parameters that identify the pure gravity Swing-By with a fixed mass ratio (angle of approach, periapsis distance and velocity at periapsis) and the three parameters that define the impulsive maneuver (direction, magnitude and the point where the impulse is applied). The mathematical model used here is the version of the restricted three-body problem that includes the Lemaître regularization, to increase the accuracy of the numerical integrations. It is developed and implemented by an algorithm that obtains the energy variation of the spacecraft with respect to the largest primary of the system in a maneuver where the impulse is applied inside the sphere of influence of the secondary body, during the passage of the spacecraft. The point of application of the impulse is a free parameter, as well as the direction of the impulse. The results make a complete map of the possibilities, including the maximum gains of energy, but also showing alternatives that can be used considering particularities of the mission.

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Acknowledgements

The authors wish to express their appreciation for the support provided by Grants # 304700/2009-6 from the National Council for Scientific and Technological Development (CNPq); Grants # 2011/08171-3 and 2014/06688-7, from São Paulo Research Foundation (FAPESP) and the financial support from the National Council for the Improvement of Higher Education (CAPES).

Author information

Authors and Affiliations

  1. Instituto Nacional de Pesquisas Espaciais, São José dos Campos, São Paulo, 12227-010, Brazil

    Alessandra F. S. Ferreira & Antônio F. B. A. Prado

  2. Universidade Estadual Paulista, Guaratinguetá, São Paulo, 12516-410, Brazil

    Othon C. Winter

Authors
  1. Alessandra F. S. Ferreira
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  2. Antônio F. B. A. Prado
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  3. Othon C. Winter
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Corresponding author

Correspondence to Alessandra F. S. Ferreira.

Appendices

Appendix 1: Detailed information about the trajectories in the Earth–Moon system

It is shown here the maximum energy variations \((\Delta E_{\max })\) and its corresponding data, as the true anomaly of the point where the impulse is applied \((\theta )\), the angle that gives the direction of the impulse \((\alpha )\), the deflection angle \((\zeta )\), the escape velocity \((V_{\infty +})\) and R, which is the distance between the spacecraft and the secondary body at the instant that the impulse is applied.

Appendix 2: Details of the trajectories for the Sun–Jupiter system

It is shown here the maximum energy variations \((\Delta E_{\max })\) and its corresponding data, as the true anomaly of the point where the impulse is applied \((\theta )\), the angle that gives the direction of the impulse \((\alpha )\), the deflection angle \((\zeta )\), the escape velocity \((V_\infty +)\) and R, which is the distance between the spacecraft and the secondary body at the instant that the impulse is applied.

Appendix 3: Coefficients of the empirical equations that describe the maximum energy variation

Equations (24) and (28) describe the coefficients of Eq. 20 as a function of the magnitude of the impulse \((\delta {V})\).

$$\begin{aligned} a_1= & {} -\left( {6.9093\times 10^{-10}} \right) \delta V^{8}+\left( {1.2491\times 10^{-8}} \right) \delta V^{7}\nonumber \\&-\,\left( {9.4632\times 10^{-8}} \right) \delta V^{6}+\left( {3.8896\times 10^{-7}} \right) \delta V^{5}\nonumber \\&-\,\left( {9.3631\times 10^{-7}} \right) \delta V^{4} +\left( {1.3226\times 10^{-6}} \right) \delta V^{3}\nonumber \\&-\,\left( {1.0262\times 10^{-6}} \right) \delta V^{2}+\left( {3.6004\times 10^{-7}} \right) \delta V\nonumber \\&-\,\left( {2.1218\times 10^{-8}} \right) \end{aligned}$$
(24)
$$\begin{aligned} b_1= & {} \left( {8.1943\times 10^{-7}} \right) \delta V^{8}-\left( {1.4749\times 10^{-5}} \right) \delta V^{7}\nonumber \\&+\,0.00011121\delta V^{6}-0.00045486\delta V^{5}\nonumber \\&+\,0.0010895\delta V^{4} -0.0015315\delta V^{3}\nonumber \\&+\,0.0011831\delta V^{2} -0.00041416\delta V\nonumber \\&+\,\left( {2.4623\times 10^{-5}} \right) \end{aligned}$$
(25)
$$\begin{aligned} c_1= & {} -0.00036168\delta V^{8}+0.0064807\delta V^{7}\nonumber \\&-\,0.048632\delta V^{6}+0.19791\delta V^{5}-0.47159\delta V^{4}\nonumber \\&+\,0.65953\delta V^{3}-0.50708\delta V^{2} +0.17702\delta V\nonumber \\&-\,0.010831 \end{aligned}$$
(26)
$$\begin{aligned} d_1= & {} 0.07033\delta V^{8}-1.2546\delta V^{7} +9.3694\delta V^{6}\nonumber \\&-\,37.935\delta V^{5} +89.917\delta V^{4}-125.08\delta V^{3}\nonumber \\&+\,95.691\delta V^{2}-33.267\delta V +2.1306 \end{aligned}$$
(27)
$$\begin{aligned} e_1= & {} -0.74231\delta V^{9}+7.6992\delta V^{8} -1.701\delta V^{7}\nonumber \\&-\,311.82\delta V^{6} +1875.7\delta V^{5}-5229\delta V^{4}\nonumber \\&+\,7892.5\delta V^{3}-6304.6\delta V^{2} +2239.9\delta V-151.11\nonumber \\ \end{aligned}$$
(28)

Equations (29) and (33) describe the coefficients of Eq. (21) as a function of the magnitude of the impulse \((\delta {V})\).

$$\begin{aligned} a_2= & {} -\left( {2.0996\times 10^{-10}} \right) \delta V^{8}+\left( {3.7802\times 10^{-9}} \right) \delta V^{7}\nonumber \\&-\,\left( {2.8519\times 10^{-8}} \right) \delta V^{6}+\left( {1.1676\times 10^{-7}} \right) \delta V^{5}\nonumber \\&-\,\left( {2.8016\times 10^{-7}} \right) \delta V^{4}+\left( {3.951\times 10^{-7}} \right) \delta V^{3}\nonumber \\&-\,\left( {3.071\times 10^{-7}} \right) \delta V^{2}+\left( {1.0911\times 10^{-7}} \right) \delta V\nonumber \\&-\,\left( {5.1707\times 10^{-9}} \right) \end{aligned}$$
(29)
$$\begin{aligned} b_2= & {} \left( {2.2047\times 10^{-7}} \right) \delta V^{8}-\left( {4.0258\times 10^{-6}} \right) \delta V^{7}\nonumber \\&+\,\left( {3.0831\times 10^{-5}} \right) \delta V^{6}-0.00012824\delta V^{5}\nonumber \\&+\,0.00031282\delta V^{4}-0.00044882\delta V^{3}\nonumber \\&+\,0.00035537\delta V^{2} -0.00012942\delta V\nonumber \\&+\,\left( {7.0061\times 10^{-6}} \right) \end{aligned}$$
(30)
$$\begin{aligned} c_2= & {} -\left( {7.2312\times 10^{-5}} \right) \delta V^{9}+0.0011587\delta V^{8}\nonumber \\&-\,0.0073561\delta V^{7}+0.022503\delta V^{6}-0.027767\delta V^{5}\nonumber \\&-\,0.019652\delta V^{4}+0.10125\delta V^{3}\nonumber \\&-\,0.11473\delta V^{2}+0.049405\delta V-0.0031216 \end{aligned}$$
(31)
$$\begin{aligned} d_2= & {} 0.016556\delta V^{9}-0.26995\delta V^{8}+1.7661\delta V^{7}\nonumber \\&-\,5.7542\delta V^{6}+8.7551-1.0826\delta V^{4}-15.562\delta V^{3}\nonumber \\&+\,20.553\delta V^{2}-9.344\delta V+0.70129 \end{aligned}$$
(32)
$$\begin{aligned} e_2= & {} -1.3364\delta V^{9}+22.003\delta V^{8}-146.32\delta V^{7}\nonumber \\&+\,492.14\delta V^{6}-817.07\delta V^{5}+345.12\delta V^{4}\nonumber \\&+\,902.47\delta V^{3}-1390.6\delta V^{2}+659.13\delta V-56.513\nonumber \\ \end{aligned}$$
(33)

Equations (34) and (44) describe the coefficients of Eq. (22) as a function of the magnitude of the impulse \((\delta {V})\).

$$\begin{aligned} a_3= & {} \left( {1.6042\times 10^{-17}} \right) \delta V^{8}-\left( {2.5231\times 10^{-16}} \right) \delta V^{7}\nonumber \\&+\,\left( {1.6116\times 10^{-15}} \right) \delta V^{6}-\left( {5.3638\times 10^{-15}} \right) \delta V^{5}\nonumber \\&+\,\left( {9.9236\times 10^{-15}} \right) \delta V^{4}-\left( {1.0104\times 10^{-14}} \right) \delta V^{3}\nonumber \\&+\,\left( {5.3072\times 10^{-15}} \right) \delta V^{2}-\left( {1.2438\times 10^{-15}} \right) \delta V\nonumber \\&+\,\left( {8.7211\times 10^{-17}} \right) \end{aligned}$$
(34)
$$\begin{aligned} b_3= & {} -\left( {4.3206\times 10^{-14}} \right) \delta V^{8}+\left( {6.7949\times 10^{-13}} \right) \delta V^{7}\nonumber \\&-\,\left( {4.3399\times 10^{-12}} \right) \delta V^{6}+\left( {1.4442\times 10^{-11}} \right) \delta V^{5}\nonumber \\&-\,\left( {2.6716\times 10^{-11}} \right) \delta V^{4}+\left( {2.72\times 10^{-11}} \right) \delta V^{3}\nonumber \\&-\,\left( {1.4287\times 10^{-11}} \right) \delta V^{2}+\left( {3.3473\times 10^{-12}} \right) \delta V\nonumber \\&-\,\left( {2.3467\times 10^{-13}} \right) \end{aligned}$$
(35)
$$\begin{aligned} c_3= & {} \left( {5.2025\times 10^{-11}} \right) \delta V^{8}-\left( {8.1812\times 10^{-10}} \right) \delta V^{7}\nonumber \\&+\,\left( {5.2248\times 10^{-9}} \right) \delta V^{6}-\left( {1.7385\times 10^{-8}} \right) \delta V^{5}\nonumber \\&+\,\left( {3.2158\times 10^{-8}} \right) \delta V^{4}-\left( {3.2737\times 10^{-8}} \right) \delta V^{3}\nonumber \\&+\,\left( {1.7197\times 10^{-8}} \right) \delta V^{2}-\left( {4.0281\times 10^{-9}} \right) \delta V\nonumber \\&+\,\left( {2.8236\times 10^{-10}} \right) \end{aligned}$$
(36)
$$\begin{aligned} d_3= & {} -\left( {3.6871\times 10^{-8}} \right) \delta V^{8}+\left( {5.7978\times 10^{-7}} \right) \delta V^{7}\nonumber \\&-\,\left( {3.7024\times 10^{-6}} \right) \delta V^{6}+\left( {1.2318\times 10^{-5}} \right) \delta V^{5}\nonumber \\&-\,\left( {2.2783\times 10^{-5}} \right) \delta V^{4}+\left( {2.3193\times 10^{-5}} \right) \delta V^{3}\nonumber \\&-\,\left( {1.2185\times 10^{-5}} \right) \delta V^{2}+\left( {2.8537\times 10^{-6}} \right) \delta V\nonumber \\&-\,\left( {2.0002\times 10^{-7}} \right) \end{aligned}$$
(37)
$$\begin{aligned} e_3= & {} \left( {1.7033\times 10^{-5}} \right) \delta V^{8}-0.00026781\delta V^{7}\nonumber \\&+\,0.00171\delta V^{6}-0.005689\delta V^{5}+0.010521\delta V^{4}\nonumber \\&-\,0.01071\delta V^{3}+0.0056279\delta V^{2}\nonumber \\&-\,0.001318\delta V+\left( {9.2376\times 10^{-5}} \right) \end{aligned}$$
(38)
$$\begin{aligned} f_3= & {} -0.0066105\delta V^{9}+0.10848\delta V^{8}\nonumber \\&-\,0.7349\delta V^{7}+2.6606\delta V^{6}\nonumber \\&-\,5.5748\delta V^{5}+6.8616\delta V^{4}\nonumber \\&-\,4.8323\delta V^{3}+1.8036\delta V^{2}\nonumber \\&-\,0.29959\delta V+0.014695 \end{aligned}$$
(39)
$$\begin{aligned} g_3= & {} 1.4437\delta V^{9}-23.699\delta V^{8}\nonumber \\&+\,160.63\delta V^{7}-581.9\delta V^{6}+1220.3\delta V^{5}\nonumber \\&-\,1503.8\delta V^{4}+1060.6\delta V^{3}\nonumber \\&-\,396.53\delta V^{2}+66.039\delta V-3.2517 \end{aligned}$$
(40)
$$\begin{aligned} h_3= & {} -214.76\delta V^{9}+3526.6\delta V^{8}-23914\delta V^{7}\nonumber \\&+\,86691\delta V^{6}-\left( {1.8196\times 10^{-5}} \right) \delta V^{5}\nonumber \\&+\,\left( {2.245\times 10^{-5}} \right) \delta V^{4}-\left( {1.5858\times 10^{-5}} \right) \delta V^{3}\nonumber \\&+\,59393\delta V^{2}-9916.8\delta +490.14 \end{aligned}$$
(41)
$$\begin{aligned} i_3= & {} 20824\delta V^{9}-\left( {3.4209\times 10^{-5}} \right) \delta V^{8}\nonumber \\&+\,\left( {2.3209\times 10^{-6}} \right) \delta V^{7}-\left( {8.4193\times 10^{-6}} \right) \delta V^{6}\nonumber \\&+\,\left( {1.7688\times 10^{-7}} \right) \delta V^{5}-\left( {2.1851\times 10^{-7}} \right) \delta V^{4}\nonumber \\&+\,\left( {1.5459\times 10^{-7}} \right) \delta V^{3}-\left( {5.8\times 10^{-6}} \right) \delta V^{2}\nonumber \\&+\,\left( {9.7087\times 10^{-5}} \right) \delta V-48163 \end{aligned}$$
(42)
$$\begin{aligned} j_3= & {} -\left( {1.1884\times 10^{-6}} \right) \delta V^{9}+\left( {1.953\times 10^{-7}} \right) \delta V^{8}\nonumber \\&-\,\left( {1.3257\times 10^{-8}} \right) \delta V^{7}+\left( {4.8126\times 10^{-8}} \right) \delta V^{6}\nonumber \\&-\,\left( {1.0121\times 10^{-9}} \right) \delta V^{5}+\left( {1.2518\times 10^{-9}} \right) \delta V^{4}\nonumber \\&-\,\left( {8.8709\times 10^{-8}} \right) \delta V^{3}+\left( {3.334\times 10^{-8}} \right) \delta V^{2}\nonumber \\&-\,\left( {5.5947\times 10^{-7}} \right) \delta V+\left( {2.7855\times 10^{-6}} \right) \end{aligned}$$
(43)
$$\begin{aligned} k_3= & {} \left( {3.0312\times 10^{-7}} \right) \delta V^{9}-\left( {4.9835\times 10^{-8}} \right) \delta V^{8}\nonumber \\&+\,\left( {3.3847\times 10^{-9}} \right) \delta V^{7}-\left( {1.2296\times 10^{-10}} \right) \delta V^{6}\nonumber \\&+\,\left( {2.5884\times 10^{-10}} \right) \delta V^{5}-\left( {3.2059\times 10^{-10}} \right) \delta V^{4}\nonumber \\&+\,\left( {2.2755\times 10^{-10}} \right) \delta V^{3}-\left( {8.5671\times 10^{-9}} \right) \delta V^{2}\nonumber \\&+\,\left( {1.4412\times 10^{-9}} \right) \delta V-\left( {7.2007\times 10^{-7}} \right) \end{aligned}$$
(44)

Equations (45) and (55) describe the coefficients of Eq. (23) as a function of the magnitude of the impulse \((\delta V)\).

$$\begin{aligned} a_4= & {} -\left( {1.0705\times 10^{-18}} \right) \delta V^{9}+\left( {1.6883\times 10^{-17}} \right) \delta V^{8}\nonumber \\&-\left( {1.1045\times 10^{-16}} \right) \delta V^{7}+\left( {3.9381\times 10^{-16}} \right) \delta V^{6}\nonumber \\&-\left( {8.5765\times 10^{-16}} \right) \delta V^{5}+\left( {1.2342\times 10^{-15}} \right) \delta V^{4}\nonumber \\&-\left( {1.229\times 10^{-15}} \right) \delta V^{3}+\left( {7.7792\times 10^{-16}} \right) \delta V^{2}\nonumber \\&-\left( {1.9375\times 10^{-16}} \right) \delta V+\left( {9.8868\times 10^{-18}} \right) \end{aligned}$$
(45)
$$\begin{aligned} b_4= & {} \left( {4.2985\times 10^{-15}} \right) \delta V^{8}-\left( {6.1705\times 10^{-14}} \right) \delta V^{7}\nonumber \\&+\left( {3.4654\times 10^{-13}} \right) \delta V^{6}-\left( {9.4239\times 10^{-13}} \right) \delta V^{5}\nonumber \\&+\left( {1.1863\times 10^{-12}} \right) \delta V^{4}-\left( {3.4127\times 10^{-13}} \right) \delta V^{3}\nonumber \\&-\left( {4.98\times 10^{-13}} \right) \delta V^{2}+\left( {1.9964\times 10^{-13}} \right) \delta V\nonumber \\&-\left( {7.101\times 10^{-15}} \right) \end{aligned}$$
(46)
$$\begin{aligned} c_4= & {} -\left( {3.1033\times 10^{-12}} \right) \delta V^{9}+\left( {4.8112\times 10^{-11}} \right) \delta V^{8}\nonumber \\&-\left( {3.0786\times 10^{-10}} \right) \delta V^{7}+\left( {1.0691\times 10^{-9}} \right) \delta V^{6}\nonumber \\&-\left( {2.2718\times 10^{-9}} \right) \delta V^{5}+\left( {3.2536\times 10^{-9}} \right) \delta V^{4}\nonumber \\&-\left( {3.3468\times 10^{-9}} \right) \delta V^{3}+\left( {2.2308\times 10^{-9}} \right) \delta V^{2}\nonumber \\&-\left( {5.6288\times 10^{-10}} \right) \delta V+\left( {2.8565\times 10^{-11}} \right) \end{aligned}$$
(47)
$$\begin{aligned} d_4= & {} \left( {3.8859\times 10^{-9}} \right) \delta V^{8}-\left( {5.6072\times 10^{-8}} \right) \delta V^{7}\nonumber \\&+\left( {3.1746\times 10^{-7}} \right) \delta V^{6}-\left( {8.761\times 10^{-7}} \right) \delta V^{5}\nonumber \\&+\left( {1.1436\times 10^{-6}} \right) \delta V^{4}-\left( {4.2105\times 10^{-7}} \right) \delta V^{3}\nonumber \\&-\left( {3.5995\times 10^{-7}} \right) \delta V^{2}+\left( {1.5266\times 10^{-7}} \right) \delta V\nonumber \\&-\left( {5.3446\times 10^{-9}} \right) \end{aligned}$$
(48)
$$\begin{aligned} e_4= & {} -\left( {8.8741\times 10^{-7}} \right) \delta V^{9}+\left( {1.3434\times 10^{-5}} \right) \delta V^{8}\nonumber \\&-(8.3238\times 10^{-5})\delta V^{7}+0.00027751\delta V^{6}\nonumber \\&-\,0.0005667\delta V^{5}+0.00080599\delta V^{4}-0.00087623\delta V^{3}\nonumber \\&+\,0.00063143\delta V^{2}-0.00016252\delta V\nonumber \\&+\left( {8.1928\times 10^{-6}} \right) \end{aligned}$$
(49)
$$\begin{aligned} f_4= & {} 0.00025791\delta V^{9}-0.0038432\delta V^{8}\nonumber \\&+\,0.023289\delta V^{7}-0.075357\delta V^{6}\nonumber \\&+\,0.14916\delta V^{5}-0.21108\delta V^{4}+0.23989\delta V^{3}\nonumber \\&-\,0.18269\delta V^{2}+0.047668\delta V-0.0023935 \end{aligned}$$
(50)
$$\begin{aligned} g_4= & {} -0.051179\delta V^{9}+0.74788\delta V^{8} -4.4036\delta V^{7}\nonumber \\&+\,13.676\delta V^{6} -25.859\delta V^{5}+36.334\delta V^{4}\nonumber \\&-\,44.11\delta V^{3}+36.119\delta V^{2} -9.5874\delta V\nonumber \\&+\,0.47923 \end{aligned}$$
(51)
$$\begin{aligned} h_4= & {} 6.8442\delta V^{9}-97.59\delta V^{8}+553.12\delta V^{7}\nonumber \\&-\,1619.3\delta V^{6}+2845\delta V^{5} -3950.4\delta V^{4}\nonumber \\&+\,5330.8\delta V^{3} -4813.3\delta V^{2}+1305.2\delta V-64.903 \end{aligned}$$
(52)
$$\begin{aligned} i_4= & {} -588.65\delta V^{9}+8132.2\delta V^{8}-43715\delta V^{7}\nonumber \\&+\left( {1.1669\times 10^{5}} \right) \delta V^{6}-\left( {1.7871\times 10^{5}} \right) \delta V^{5}\nonumber \\&+\left( {2.4225\times 10^{5}} \right) \delta V^{4}-\left( {3.9806\times 10^{5}} \right) \delta V^{3}\nonumber \\&+\left( {4.1292\times 10^{5}} \right) \delta V^{2}-\left( {1.15\times 10^{5}} \right) \delta V+5684.5 \nonumber \\\end{aligned}$$
(53)
$$\begin{aligned} j_4= & {} 29286\delta V^{9}-\left( {3.8778\times 10^{5}} \right) \delta V^{8}\nonumber \\&+\left( {1.9269\times 10^{6}} \right) \delta V^{7}-\left( {4.355\times 10^{6}} \right) \delta V^{6}\nonumber \\&+\left( {4.6593\times 10^{6}} \right) \delta V^{5}-\left( {5.8182\times 10^{6}} \right) \delta V^{4}\nonumber \\&+\left( {1.5982\times 10^{7}} \right) \delta V^{3}-\left( {2.053\times 10^{7}} \right) \delta V^{2}\nonumber \\&+\left( {5.9161\times 10^{6}} \right) \delta V-\left( {2.9038\times 10^{5}} \right) \end{aligned}$$
(54)
$$\begin{aligned} k_4= & {} -\left( {6.389\times 10^{5}} \right) \delta V^{9}+\left( {7.9757\times 10^{6}} \right) \delta V^{8}\nonumber \\&-\left( {3.4901\times 10^{7}} \right) \delta V^{7}+\left( {5.3272\times 10^{7}} \right) \delta V^{6}\nonumber \\&+\left( {2.0075\times 10^{7}} \right) \delta V^{5}-\left( {5.2271\times 10^{7}} \right) \delta V^{4}\nonumber \\&-\left( {2.3901\times 10^{8}} \right) \delta V^{3}+\left( {4.4785\times 10^{8}} \right) \delta V^{2}\nonumber \\&-\left( {1.3483\times 10^{8}} \right) \delta V+\left( {6.5623\times 10^{6}} \right) \end{aligned}$$
(55)

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Ferreira, A.F.S., Prado, A.F.B.A. & Winter, O.C. A numerical mapping of energy gains in a powered Swing-By maneuver. Nonlinear Dyn 89, 791–818 (2017). https://doi.org/10.1007/s11071-017-3485-2

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