^{1}

The combination of atmospheric drag and lunar and solar perturbations in addition to Earth’s oblateness influences the orbital lifetime of an upper stage in geostationary transfer orbit (GTO). These high eccentric orbits undergo fluctuations in both perturbations and velocity and are very sensitive to the initial conditions. The main objective of this paper is to predict the reentry time of the upper stage of the Indian geosynchronous satellite launch vehicle, GSLV-D5, which inserted the satellite GSAT-14 into a GTO on January 05, 2014, with mean perigee and apogee altitudes of 170 km and 35975 km. Four intervals of near linear variation of the mean apogee altitude observed were used in predicting the orbital lifetime. For these four intervals, optimal values of the initial osculating eccentricity and ballistic coefficient for matching the mean apogee altitudes were estimated with the response surface methodology using a genetic algorithm. It was found that the orbital lifetime from these four time spans was between 144 and 148 days.

The estimation of the orbital lifetime of decaying upper stages is important, due to the impact risks associated. In fact, the geostationary transfer orbit (GTO) is a high eccentric orbit which traverses low Earth orbit (LEO) and geostationary orbit (GSO), where the threat of collision with an operational satellite is high. Hence, minimizing the lifetime of an upper stage is a significant consideration in space debris mitigation endeavors. In order to reduce the collision probability, GSO satellites are placed in graveyard orbits at the end of their operational life.

Lunar and solar perturbations on geostationary transfer orbits account for the oscillation of the eccentricity keeping the semimajor axis constant [

In this paper, a method to perform the lifetime estimation of an upper stage in GTO is presented as an optimal estimation problem [

The orbital elements data given as two-line element sets (TLEs) were downloaded from the Space Track website from February 11, 2014, to April 4, 2014. Since the orbital period was greater than 225 minutes, the SDP4 orbit propagator model [

The response surface methodology (RSM) is a combination of mathematical and statistical techniques used widely in various fields for the purpose of development of models and optimization [

The actual relationship between the response and the independent variables is usually unknown. Therefore, the first step in RSM is to find the approximate model function which shows the true relationship. Generally, the approximation is started with a low-order polynomial function. If the response is characterized by a linear function, then a first-order model is used as approximation function. If the response surface has a curvature, then a higher-order polynomial function is used, such as a second-order model. The objective of RSM is not only to understand the contour of the response surface, but also to estimate the values of independent variables called “design variables,” for which the optimal response is obtained.

In order to obtain the optimal response, a genetic algorithm (GA) which belongs to the group of evolutionary algorithms (EA) is used [

Creation of an initial population: an initial set of solutions or chromosomes is generated randomly or by a seeding procedure.

Fitness evaluation: the quality of each solution of the population is evaluated using the fitness function.

Selection: with this process promising solutions are chosen to pass to the next generation at the expense of other solutions which are considered ill-equipped for the objective.

Crossover: this process consists of taking two strings or parents from the population and performing a random exchange of portions between them to form a new solution. This new chromosome has information from both parents. The crossover does not apply to the entire population string, but it is limited by the crossover rate.

Mutation: this involves making changes in individual values of variables in a solution. Mutations serve to maintain the diversity of the population, reducing the probability of finding a local minimum or local maximum rather than the global optimal solution.

Checking if the stopping criterion is satisfied: if the stopping criterion is not satisfied, the process returns to step number 3. If the criterion is satisfied, the algorithm finishes.

For the reentry time prediction, we consider the initial osculating eccentricity

Cycle of genetic algorithm.

Variation of observed mean apogee altitude of GSLV R/B.

The TLEs downloaded from the Space Track Organization website are converted into state vectors using the SatSpy software and the mean and osculating orbital elements are computed using the NPOE software. The observed mean apogee and perigee altitudes computed from the TLEs are plotted in Figures

Variation of observed mean perigee altitude of GSLV R/B.

The osculating orbital elements of GSLV R/B (NORAD No. 39499) as obtained from the TLE on February 13, 2014, 15 : 24 (UTC), are as follows:

semimajor axis (km) = 24163.152291,

eccentricity = 0.7279715192,

inclination (°) = 19.33662306,

argument of perigee (°) = 207.14804219,

right ascension of the ascending node (°) = 196.09051738,

true anomaly (°) = 152.63951389.

To generate a set of mean apogee surfaces for zone A, three values of initial osculating eccentricity (0.7278715192, 0.7279715192, and 0.7280715192) and three values of ballistic coefficient (60, 80, and 100) are selected to obtain nine grid points as plotted in Figure

Observed and predicted mean apogee altitude for zone A.

TLE epoch (UTC) | Mean apogee observed (km) | Mean apogee predicted (km) | Error (km) |
---|---|---|---|

February 13, 2014, 15:24 | 35383.75000 | 35383.56640 | 0.184 |

February 14, 2014, 12:08 | 35378.30469 | 35379.12109 | −0.816 |

February 15, 2014, 19:12 | 35372.59375 | 35373.03125 | −0.438 |

February 16, 2014, 05:33 | 35371.17969 | 35371.17578 | 0.004 |

February 17, 2014, 22:59 | 35364.45313 | 35363.92578 | 0.527 |

February 18, 2014, 19:41 | 35357.75781 | 35360.13672 | −2.379 |

February 19, 2014, 06:02 | 35357.01172 | 35357.91797 | −0.906 |

February 20, 2014, 13:05 | 35348.90234 | 35350.23828 | −1.336 |

Set of mean apogee surfaces for zone A with (

The number of chromosomes is 24 and the mating pool is

Mate =

A number of crossovers and mutations were performed on the chromosomes to get the best solution. The fitness or the quality of the solution is determined by

The osculating orbital elements of GSLV R/B as obtained from the TLE on March 2, 2014, 10 : 45 (UTC), are as follows:

semimajor axis (km) = 24087.380496,

eccentricity = 0.7279210409,

inclination (°) = 19.33807553,

argument of perigee (°) = 219.63775080,

right ascension of the ascending node (°) = 189.22925587,

true anomaly (°) = 140.11376942.

To generate a set of mean apogee surfaces for zone B, three values of initial osculating eccentricity (0.7278210409, 0.7279210409, and 0.7280210409) and three values of ballistic coefficient (50, 80, and 110) are selected to obtain nine grid points as plotted in Figure

Set of mean apogee surfaces for zone B with (

With the initial estimates of

Observed and predicted mean apogee altitude for zone B.

TLE epoch (UTC) | Mean apogee observed (km) | Mean apogee predicted (km) | Error (km) |
---|---|---|---|

March 2, 2014, 10:45 | 35250.6328125 | 35250.03125 | 0.602 |

March 3, 2014, 17:43 | 35240.3906300 | 35240.16016 | 0.230 |

March 4, 2014, 04:01 | 35230.7539100 | 35236.41406 | −5.660 |

March 5, 2014, 10:58 | 35222.8984400 | 35223.78516 | −0.887 |

March 6, 2014, 17:53 | 35217.1718800 | 35209.39844 | 7.773 |

March 7, 2014, 04:11 | 35204.4765600 | 35204.16797 | 0.309 |

The osculating orbital elements of GSLV R/B as obtained from the TLE on March 10, 2014, 14 : 32 (UTC), are as follows:

semimajor axis (km) = 24031.141029,

eccentricity = 0.7277057104,

inclination (°) = 19.33698821,

argument of perigee (°) = 225.83231865,

right ascension of the ascending node (°) = 185.82625378,

true anomaly (°) = 133.79218860.

To generate a set of mean apogee surfaces for zone C, three values of initial osculating eccentricity (0.7276057104, 0.7277057104, and 0.7278057104) and three values of ballistic coefficient (50, 80, and 110) are selected to obtain nine grid points as plotted in Figure

Set of mean apogee surfaces for zone C with (

With the initial estimates of

Observed and predicted mean apogee altitude for zone C.

TLE epoch (UTC) | Mean apogee observed (km) | Mean apogee predicted (km) | Error (km) |
---|---|---|---|

March 10, 2014, 14:32 | 35147.26563 | 35145.14063 | 2.125 |

March 11, 2014, 11:07 | 35135.21484 | 35133.85938 | 1.355 |

March 12, 2014, 07:41 | 35123.41797 | 35122.85547 | 0.563 |

March 12, 2014, 17:57 | 35118.95313 | 35117.47266 | 1.480 |

March 13, 2014, 14:31 | 35105.96094 | 35106.90625 | −0.945 |

March 14, 2014, 00:47 | 35100.91406 | 35101.61328 | −0.699 |

March 15, 2014, 07:35 | 35082.73828 | 35086.21484 | −3.477 |

March 17, 2014, 21:07 | 35052.68750 | 35052.47656 | 0.211 |

The osculating orbital elements of GSLV R/B as obtained from the TLE on March 24, 2014, 06 : 31 (UTC), are as follows:

semimajor axis (km) = 23888.894645,

eccentricity = 0.7268822734,

inclination (°) = 19.29960868,

argument of perigee (°) = 236.26578322,

right ascension of the ascending node (°) = 180.05835079,

true anomaly (°) = 121.02982099,

To generate a set of mean apogee surfaces for zone D, three values of initial osculating eccentricity (0.7267822734, 0.7268822734, and 0.7269822734) and three values of ballistic coefficient (50, 80, and 110) are selected to obtain nine grid points as plotted in Figure

Observed and predicted mean apogee altitude for zone D.

TLE epoch (UTC) | Mean apogee observed (km) | Mean apogee predicted (km) | Error (km) |
---|---|---|---|

March 24, 2014, 06:31 | 34877.94531 | 34876.13281 | 1.813 |

March 25, 2014, 02:58 | 34839.96094 | 34850.94141 | −10.980 |

March 26, 2014, 19:43 | 34801.82031 | 34802.17969 | −0.359 |

March 29, 2014, 08:41 | 34723.11328 | 34727.16016 | −4.047 |

March 31, 2014, 01:15 | 34665.50781 | 34665.56250 | −0.055 |

Set of mean apogee surfaces for zone D with (

From the above four zones, the orbital lifetime of GSLV-D5 is found to be between 144 and 148 days, which is a small variation between the four lifetime values from four different epochs. The near linear variation of mean apogee altitude has shown the reentry time more accurately from the TLEs considered. Hence, it is proven once again that the method based on near linear variation of mean apogee altitude utilized in [

Parameters of the genetic algorithm.

Zone label | Number of chromosomes | Number of GA generations to converge | Crossover probability | Mutation probability |
---|---|---|---|---|

A | 24 | 15 | 0.8 | 0.01 |

B | 32 | 9 | 0.8 | 0.01 |

C | 36 | 23 | 0.8 | 0.01 |

D | 34 | 18 | 0.8 | 0.01 |

Computed values of initial osculating eccentricity and ballistic coefficient and reentry time for each zone using RSM with GA.

Zone label | TLEs considered (UTC) | Computed values | Predicted reentry time (mm-dd-yyyy) |
Time interval^{a} (days) | ||
---|---|---|---|---|---|---|

From | To | Initial osculating eccentricity ( |
Initial ballistic coefficient ( |
|||

A | February 13, 2014, 15:24 | February 20, 2014, 13:05 | 0.72796 | 84.4065933 | June 1, 2014 | 147 |

B | March 2, 2014, 10:45 | March 7, 2014, 04:11 | 0.72789621 | 85.8965302 | June 1, 2014 | 147 |

C | March 10, 2014, 14:32 | March 17, 2014, 21:07 | 0.72761744 | 81.6301346 | June 2, 2014 | 148 |

D | March 24, 2014, 06:31 | March 31, 2014, 01:15 | 0.72680628 | 76.4813202 | May 29, 2014 | 144 |

The justification for using RSM to predict mean apogee altitude, as opposed to using NPOE software to propagate the trajectory forward (starting with the initial values of

The response surface technique with the genetic algorithm is utilized to obtain the optimal values of the initial osculating eccentricity and the ballistic coefficient of each of the selected time intervals based on the near linear variation of mean apogee altitude. Using these optimal values, the orbital lifetime of a GSLV-D5 rocket body is found to be between 144 and 148 days from its injection into the orbit on January 5, 2014.

Geostationary transfer orbit

Geosynchronous satellite launch vehicle

Low Earth orbit

Geostationary orbit

Drag coefficient

Mass of the object

Effective area

Rocket body

Two-line element

Genetic algorithm

Numerical prediction of orbital events

Simplified general perturbations

Simplified deep space perturbations

Solar flux

Geomagnetic index

Response surface methodology

Independent variables

Random error

Response

Evolutionary algorithms

Osculating eccentricity

Ballistic coefficient

Four zones

Zonal and tesseral harmonic terms

Goddard Earth model 10B

Mass-Spectrometer-Incoherent-Scatter-1990 atmosphere model

North American Aerospace Defense Command

Coordinated universal time

Three values of osculating eccentricity

Three values of ballistic coefficient

First value of osculating eccentricity and ballistic coefficient

First value of osculating eccentricity and second value of ballistic coefficient

Average dispersions in apogee

Observed apogee altitude

Apogee surface

Time instants for observations

Number of observations

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are thankful to Dr. T. V. Christy, Director of School of Mechanical Engineering, and Dr. Pradeep Kumar, Head of Aerospace Department, for their constant support and motivation.