Orbit, orbital lifetime, and reentry survivability estimation for orbiting objects

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ScienceDirect Advances in Space Research xxx (2018) xxx–xxx www.elsevier.com/locate/asr

Orbit, orbital lifetime, and reentry survivability estimation for orbiting objects Seong-Hyeon Park a, Hae-Dong Kim b, Gisu Park a,⇑ a

Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea b Korea Aerospace Research Institute, Daejeon 305-333, Republic of Korea

Received 1 February 2018; received in revised form 30 July 2018; accepted 9 August 2018

Abstract In this work, an integrated system for the orbit, orbital lifetime, and reentry survivability estimation modules of orbiting objects has been proposed and developed. Each module of the system was compared and validated with results from the well-known existing codes. One practical test model was considered for the orbital lifetime and the reentry survivability. The model was Science and Technology Satellite-3 (STSAT-3). Recently, the STSAT-3 was almost to collide with a space debris. This issue brought a serious alert to the public in Korea regarding collision risks of the orbiting object and a possibility of a consequent threat to the survivability when the object survives and impacts human. For this case study, it was assumed that the satellite consists of 12 parts having different shapes, materials, and sizes. The estimations showed that the calculated orbital lifetime was about 32 years and 7 out of 12 parts survived during the reentry. The effect of true anomaly for the orbiting object on the reentry survivability has been considered in the calculation. The results showed that there is no strong effect on the survivability for different true anomalies in a circular orbit. For an elliptic orbit which is regarded as an extreme case for the near-Earth orbiting object, however, a large difference was observed. Possible reasons are discussed. This work is intended to contribute to the part of the development of the national Space Situational Awareness (SSA) system, for the first time showing an overall procedure for the method of the integrated system. Ó 2018 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Orbit prediction; Orbital lifetime; Reentry survivability

1. Introduction Space debris has been increasing in number ever since the first satellite Sputnik I was launched by the Soviet Union in 1957. In particular, the number has been increased dramatically due to a collision between the active American satellite Iridium 33 and the inactive Russian communications satellite Cosmos 2251 on February 10, 2009, as well as the intentional destruction of the Fengyun-1C satellite on January 11, 2007 (Wang, 2010).

⇑ Corresponding author.

E-mail addresses: [email protected] (S.-H. Park), [email protected] (H.-D. Kim), [email protected] (G. Park).

In the Low Earth Orbit (LEO), a majority of the space debris exist and they have orbital velocity of more than 7 km/s. Thus, they can produce a high kinetic energy so that a collision of even small space debris against other objects such as satellites or a space station not only causes a disaster but produces a large number of debris (Sato, 1999). The Space Surveillance Network (SSN) of United States Strategic Command (USSTRATCOM) tracks the orbiting objects by operating radars and optical telescopes to determine their orbits and other characteristic parameters (Mehrholz et al., 2002). The USSTRATCOM provides data to the public, in which the data has a format of a Two-Line Elements set (TLEs) that encodes a list of elements of the orbiting objects at epoch time. Using the

https://doi.org/10.1016/j.asr.2018.08.016 0273-1177/Ó 2018 COSPAR. Published by Elsevier Ltd. All rights reserved.

Please cite this article in press as: Park, S.-H., et al. Orbit, orbital lifetime, and reentry survivability estimation for orbiting objects. Adv. Space Res. (2018), https://doi.org/10.1016/j.asr.2018.08.016

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TLE, one may predict the object’s position and velocity at any point in the past or future. GPS data is another option to obtain the orbital information, but the TLE is widely used and available for space debris and nanosatellites (Dell’Elce et al., 2015). Lifetime of the orbiting objects in the LEO is limited due to perturbations such as solar radiation pressure and atmospheric drag. The objects will eventually reenter the Earth’s atmosphere (Lips and Fritsche, 2005). Research on orbit prediction including lifetime has been investigated for several decades. Table 1 summarizes the research that has been performed. To obtain the position and velocity of an orbiting object, three well-known techniques (general perturbation, special perturbation, and semi-analytic) have been considered. General perturbation techniques provide an analytical solution based on the application of perturbation theories (San-Juan et al., 2017). Although the accuracy can be reduced because the most relevant forces are taken into account and the resulting expressions are truncated, they have high computational efficiency. Special perturbation techniques numerically integrate the equations of motion including complex perturbation models. Unlike the general perturbation techniques, they give accurate results but low computational efficiency. Semi-analytical techniques combine the main features of both the general and the special perturbation techniques (San-Juan et al., 2016, 2017). The equations of motion including any complex perturbing effects can be simplified using analytical techniques. Then, the transformed equations of motion can be integrated numerically through longer integration steps. As a result, the semi-analytical

techniques are suited for long-term prediction such as lifetime estimation. Recently, the hybrid perturbation theory or the probabilistic assessment method of lifetime have been proposed to increase the accuracy of orbit prediction (Dell’Elce and Kerschen, 2015; Dell’Elce et al., 2015; San-Juan et al., 2016, 2017). The hybrid perturbation theory combines an integration technique and a prediction technique based on statistical time series models or computational intelligence methods (San-Juan et al., 2016). On the other hand, the probabilistic assessment method takes into account dominant sources of parametric uncertainties and modeling errors of orbital lifetime estimation of LEO objects. The uncertainties in the initial state of the object, atmospheric drag force, and physical properties of the object are considered (Dell’Elce et al., 2015). Although the two proposed methods can provide precise results, they have some complexity and computational burden compared to the semi-analytical techniques. The semianalytical techniques still have been widely used in recent studies, and the results have shown good agreement with observed data (Afful, 2013; Le Fe`vre et al., 2014; Bernard et al., 2015; Dutt and Anilkumar, 2017). Most of reentry objects at hypervelocity speeds undergo ablation and/or melting due to the aerodynamic heating and consequently are burned out in the atmosphere. Fragments that survive may cause damage or injure people. Space debris mitigation standards or handbooks have been established by various national organizations of the space faring nations to specify an upper limit of the acceptable risk (Kato, 2001). To verify whether the space debris or

Table 1 Literature survey of orbit prediction research. References

Technique

Assessment

Description

Sterne (1958) Brouwer (1959) Brouwer and Hori (1961) Lane et al. (1962) Lane and Cranford (1969) Pimm (1971) Cefola et al. (1974) Lane and Hoots (1979) Hoots and Roehrich (1980) Liu and Alford (1980) Hoots and France (1987) Kinoshita and Nakai (1988) Chao and Platt (1991) Berry and Healy (2004) Park and Scheeres (2006) Giza et al. (2009) Cefola et al. (2009) Fujimoto et al. (2012) Afful (2013) Dell’Elce et al. (2015) San-Juan et al. (2016) Dutt and Anilkumar (2017) San-Juan et al. (2017)

General General General General General Semi-analytic Semi-analytic General General Semi-analytic General Special Semi-analytic Special – – Semi-analytic – Semi-analytic – Hybrid Semi-analytic Hybrid

Deterministic Deterministic Deterministic Deterministic Deterministic Deterministic Deterministic Deterministic Deterministic Deterministic Deterministic Deterministic Deterministic Deterministic Probabilistic Probabilistic Deterministic Probabilistic Deterministic Probabilistic Deterministic Deterministic Deterministic

Lifetime estimation Gravitational model Atmospheric drag Atmospheric model SGP4 model Lifetime estimation Lifetime estimation SGP4 model SGP models Lifetime estimation Orbit prediction Orbit prediction Lifetime estimation Orbit prediction Uncertainty quantification Uncertainty quantification Lifetime estimation Uncertainty quantification Lifetime estimation Uncertainty quantification Orbit prediction Lifetime estimation HSGP4 model

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the satellites comply with the applicable standards, reentry analysis codes have been developed by a number of space agencies and research centers (Lips and Fritsche, 2005; Sim and Kim, 2011; Lee et al., 2016). Table 2 summarizes the representative reentry analysis codes. The reentry analysis codes can be classified into two categories: object-oriented code and spacecraft-oriented code (Lips and Fritsche, 2005; Wu et al., 2011). The Object Reentry Survival Analysis Tool (ORSAT) by NASA and Spacecraft Atmospheric Reentry and Aerothermal Breakup (SCARAB) by HTG are the most representative object-oriented and spacecraft-oriented codes, respectively (Sim and Kim, 2011; Choi et al., 2017). Although they use different approaches for various analysis such as trajectory and heating rates, both the codes have shown good agreement with simple shapes (Lips et al., 2005). Most of the reentry analysis codes are not open to the public, but there are a few open codes such as Debris Assessment Software (DAS) by NASA and Debris Risk Assessment and Mitigation Analysis (DRAMA) by ESA. Even though they can be easily accessible and available, they have some limitations compared to the ORSAT and the SCARAB. To name a few, the DAS uses a lumped thermal mass model and temperature independent material properties. Consequently, the DAS is not able to predict partial melting and accurate heat rates. The DRAMA also has similar limitations to that of DAS in which the oxidation heating rate is not considered in an aerothermal analysis (Lips and Fritsche, 2005). In addition, the output of the analysis is limited to mass, cross-section, velocity, incident angle, and impact location, so detailed results such as heating rates cannot be confirmed (Lips and Fritsche, 2005). For this reason, a self-development is required for the reentry analysis. In practice, the reentry problem is quite complicated and there are a lot of uncertainties which may affect casualty area. In light of this, probabilistic approaches have been proposed to calculate the uncertainties in casualty area (Frank et al., 2005; Wu et al., 2011). Initial conditions, material properties, and model parameters are considered as the uncertainties and thousands of simulations such as direct simulation Monte Carlo (DSMC) method are performed to account for the uncertainties (Frank et al., 2005; Wu et al., 2011). Although the probabilistic approaches can be more appropriate than the deterministic ones, they have a computational burden. Furthermore,

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because the probabilistic approaches should be implemented based on precise deterministic approaches, the ORSAT and the SCARAB still prefer the deterministic ones. Recently, Tiangong-1 space lab, China’s first space station, reentered and burned up in the atmosphere over the southern Pacific Ocean on April 1. This issue brought a serious alert to the public in connection with ground risk and increased interest in space debris all over the world. Since an increase in the number of space debris poses a continual threat to the Earth, it is critical to collect their data and prepare for the risk. The scope of this research is to show an overall procedure of the method of developing the integrated system for the orbit, orbital lifetime, and reentry survivability estimation of orbiting objects. Because the integrated system requires a reasonable level of computational power as well as accuracy, simplified perturbation models and semi-analytic theory are used for the orbit and orbital lifetime estimations, respectively. For the reentry analysis, the developed code based on the ORSAT is applied using thermochemical equilibrium. There is a national Space Situational Awareness (SSA) system in relation to the integrated system. The SSA system has been proposed by several space agencies to understand and predict the physical location of the orbiting objects around the Earth and the possible threats to space assets, as well as mitigate or reduce the hazards (Kennewell and Vo, 2013). Korea Astronomy and Space Science Institute (KASI) is also currently developing the SSA system, which includes the Integrated Analysis System (IAS) and Space objects Monitoring System (SMS), for the threat of large artificial satellites and asteroids that crash the Earth and the prediction of potential collisions between the space objects (Choi et al., 2017). Regarding the IAS system, an analysis of the reentry prediction for the Tiangong-1 space lab has been conducted with the help of the open codes such as Semi-analytic Tool for End of Life Analysis (STELA) and the DRAMA (Choi et al., 2017). However, the analysis using these open codes has various limitations including aforementioned aspects. Moreover, the algorithms of these open codes are not open source, so they cannot be easily modified and improved. To date, to the authors’ understanding, the orbit, orbital lifetime, and reentry survivability estimation of the orbiting objects in a form of the integrated system with an overall procedure for the method within a single manuscript, has

Table 2 Summary of the representative reentry codes. Name

Country

Category

Assessment

References

SCARAB DRAMA ORSAT DAS SAPAR DRAPS ORSAT-J

Europe Europe United States United States South Korea China Japan

Spacecraft-oriented Object-oriented Object-oriented Object-oriented Object-oriented Object-oriented Object-oriented

Deterministic Deterministic Deterministic Deterministic Deterministic Probabilistic Deterministic

Lips et al. (2004) Lips and Fritsche (2005) Rochelle et al. (1997) Lips and Fritsche (2005) Sim and Kim (2011) Wu et al. (2011) Lips and Fritsche (2005)

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not yet been documented. In addition, the effect of true anomaly along a circular orbit and/or an elliptic orbit for the orbiting objects on the reentry survivability has not been investigated. This work intends to contribute to the part of the development of the national SSA system as well as to the relevant international community, which serves as the motivation for the present development. 2. Integrated system Integrated system consists of the orbit prediction, the orbital lifetime analysis, and the reentry survivability analysis modules. Fig. 1 shows a block diagram of the integrated system. Detailed information regarding each module will be discussed later. Therefore, simple explanations regarding the system are as follows. General perturbation element sets for all the space objects in Earth orbits are maintained by the North American Aerospace Defense Command (NORAD) and the USSTRATCOM. These element sets are periodically refined to retain a reasonable prediction ability for the objects and they are published as a form of Two-Line Elements (TLE) set (Hoots and Roehrich, 1980). The TLEs contain parameters that stand for the state of orbiting objects including mean orbital elements (inclination, right ascension of the ascending node (RAAN), eccentricity, argument of perigee, mean anomaly, and mean motion) and identifier information such as satellite and launch numbers (Coffee et al., 2013). In the orbit prediction module (Module-1), simplified perturbation models are used along with the TLEs to estimate position and velocity of the satellite or space debris in the True Equator Mean Equinox (TEME) reference frame because the TLEs contain the mean orbital elements obtained by removing periodic variations (Hoots and Roehrich, 1980). The simplified perturbation models consisted of SGP, SGP4, SDP4, SGP8, and SDP8 models.

While Simplified General Perturbation models (SGP, SGP4, and SGP8) are applicable to the near-Earth objects with an orbital period of less than 225 min, Simplified Deep Space Perturbations models (SDP4 and SDP8) are applicable to the objects with an orbital period greater than 225 min. In the orbital lifetime analysis module (Module-2), the orbital lifetime for the near-Earth objects perturbed by the atmospheric drag and Earth oblateness is estimated using the TLEs, and orbital elements in the TEME frame are calculated according to the lifetime. The orbital elements at the end of the lifetime can be used as an initial condition for the reentry survivability analysis. The transformation from the classical orbital element set to the spherical (ADBARV) coordinate system based on the Earth-Centered Earth-Fixed (ECEF) or Earth-Centered Inertial (ECI) reference frames is made for this (Chobotov, 1996). The survivability module (Module-3) is then used to analyze the reentry trajectory and survivability of the orbiting objects through a proper modeling process (e.g., shape, size, material, etc.) to be discussed in detail in the following section. As mentioned, the Module-1 in the figure predicts the position and velocity of the orbiting object. The simplified models used in this module are well known to predict osculating orbits from the TLEs. However, these models are not suitable for a long-term prediction because a constant ballistic coefficient is used in the drag term and the effect of drag on the density model is neglected (Vallado, 1997). And the approach usually requires a long time to calculate to capture the behaviour of the osculating orbits. For these reasons, the Module-2 is considered to estimate a decay orbital lifetime of the low-altitude object. For a long-term prediction (lifetime estimation) of the near-Earth objects, perturbations by atmospheric drag and Earth oblateness are the most dominant and important (Liu and Alford, 1980). In the Module-2, the semi-analytic

Fig. 1. Block diagram of the integrated system.

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theory is used to consider the perturbations. This approach is time-wise economic in calculation and a non-constant ballistic coefficient is used. To calculate the orbital lifetime followed by the reentry survivability, the combination of the Modules-1 and 3 can be used with the TLEs when the orbital lifetime ends less than a few days. However, in the case of the near-Earth objects have the orbital lifetime much longer than months, in the order of a few decades, the combination of the Modules-2 and 3 is a preferable option because the semi-analytic theory is suitable for a long-term prediction (Dutt and Anilkumar, 2017). 2.1. Orbit prediction The simplified perturbation models are used to predict the object’s osculating orbit. To estimate the object’s orbit such as space debris, satellite, and planet, an orbit determination is conducted using orbital dynamics model, initial orbit information, and observation data. From the determination, the NORAD and the USSTRATCOM periodically provide the general perturbation element sets to the public in the form of TLEs. To obtain a good orbit prediction from the TLEs, the simplified perturbations models such as SGP, SGP4, SDP4, SGP8, and SDP8 are generally used. The models predict the effect of perturbations caused by the Earth’s shape, radiation, and gravitational effect from other planets. Depending on the period of orbiting objects, they are classified as near-Earth objects (orbital period less than 225 min) or deep-space objects (orbital period greater than 225 min) (Hoots and Roehrich, 1980). The Simplified General Perturbation (SGP) models and the Simplified Deep Space Perturbations (SDP) models are applicable to near-Earth objects and deep-space objects, respectively. The SGP4 and SDP4 models are mainly used in the simplified perturbations models. The SGP4 is one of the analytical orbit models that could be applied for the orbit prediction using the TLE. It has been used for nearEarth satellites (Lane and Hoots, 1979; Aida and Kirschner, 2013). This model was developed by Cranford in 1970, obtained by simplifying an extensive analytical theory of Lane and Cranford (Lane and Cranford, 1969; Lane and Hoots, 1979). Contrary to the SGP4, the SDP4 is used for the deep-space objects, and it was developed by Hujsak (1979). Recently, the hybrid SGP4 model, which comprises the standard SGP4 model and an error corrector, has been proposed to extend the TLE validity because the accuracy of TLE propagations through the SGP4 model decreases over time (San-Juan et al., 2017). However, the hybrid SGP4 model requires some additional information as well as computational burden to be applied to the integrated system, so the SGP4 model is used in this study. Simple explanations of the SGP4 are given below. Detailed information and equations of the other simplified perturbation models can be found in Hoots and Roehrich (1980).

5

n0 1 þ d0 a0 a000 ¼ 1  d0
 1 134 3 d1 ; a0 ¼ a1 1  d1  d21  3 81

n000 ¼

ð1Þ ð2Þ
23 ke a1 ¼ no

3 k 2 ð3 cos2 i0  1Þ 3 k 2 ð3 cos2 i0  1Þ ; d ¼ 1 3 2 a20 ð1  e2 Þ2 2 a21 ð1  e2 Þ32 0 0 pffiffiffiffiffiffiffiffi 1 k e ¼ GM ; k 2 ¼ J 2 a2E 2 d0 ¼

ð3Þ ð4Þ ð5Þ

where the subscript 0 indicates the mean elements. The symbol n000 and a000 are original mean motion and semimajor axis, respectively. Since the orbit elements except for mean motion from the TLE are the mean quantities defined by Brouwer, the mean motion is converted from Kozai mean motion to Brouwer mean motion for the first step (Hoots et al., 2004). n is mean motion, i is inclination, e is eccentricity, G is Newton’s universal gravitational constant, M is mass of the Earth, J 2 is the second gravitational zonal harmonic of the Earth, and aE is equatorial radius of the Earth. Many constants needed for calculation can be obtained using the orbit elements from the TLE. With the constants, the secular effects of atmospheric drag and gravitation, the long-period periodic terms as well as the short-period periodic terms are considered. Then, unit orientation vectors are calculated from the following equations, U ¼ M sin uk þ N cos uk ;

V ¼ M cos uk  N sin uk

where 2

3 M x ¼  sin Xk cos ik 6 7 M ¼ 4 M y ¼ cos Xk cos ik 5; M z ¼ sin ik

ð6Þ

2

3 N x ¼ cos Xk 6 7 N ¼ 4 N y ¼ sin Xk 5 Nz ¼ 0

The subscript k denotes the osculating variables and X is longitude of ascending node. Finally, the position and velocity can be obtained, r ¼ rk U ;

v ¼ r_ k U þ ðrf_ Þk V

ð7Þ

where the equations for uk ; rk ; r_ k , and rf_ k can be found in Hoots and Roehrich (1980). 2.2. Orbital lifetime analysis Analytic solutions which consider the effects of atmospheric drag on the motion of the orbiting objects have been suggested by diverse researchers (Brouwer and Hori, 1961; Chen, 1974; Mueller, 1977). However, these solutions are usually based on the simplified analytical models related to atmospheric density, and have some limitations on perigee height due to the use of series expansions and/ or the size of eccentricity (Liu and Alford, 1980). Hence, to improve the efficiency through the use of analytical techniques and to consider the effect of atmospheric drag for

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the gravitational field of an oblate Earth, an alternative approach named ‘semi-analytic method’ has been developed (Kaufman and Dasenbrock, 1972). It uses a combination of general and special perturbation techniques. The semi-analytic theory has been used and investigated by various researchers. Chao and Platt have developed a LIFETIME tool which predicts an orbital lifetime of LEO satellite based on the semi-analytic Liu theory (Chao and Platt, 1991). To improve the accuracy of orbital lifetime prediction, this tool uses differential correction for the satellite ballistic coefficient estimation from orbit decay data. Afful has analyzed the orbital lifetime of LEO nanosatellites using the semi-analytic Liu theory (Afful, 2013). The accelerating orbit decay by deorbitsail was investigated in the analysis. Recently, the lifetime estimation has been conducted by Dutt and Anilkumar (2017). They used a BFGS Quasi-Newton algorithm to minimize least square error of perigee and apogee altitudes for the ballistic coefficient. The semi-analytic equation of motion by Chao and Platt was used and the results showed good agreement with the observed lifetime data. In this study, the semi-analytic Liu theory is used to analyze the orbital lifetime for the near-Earth objects perturbed by the atmospheric drag and the Earth oblateness. Concerning the low altitude orbiting objects with small eccentricity, external forces such as solar radiation pressure, solar and lunar gravitational effects, and higher order geopotential perturbations were neglected because the drag effect from the atmospheric density is usually larger than the effects caused by the forces (Liu and Alford, 1980). Although this theory is limited to the low altitude objects, it has a good computational efficiency to be used in the integrated system. The averaged equations of motion for the near-Earth objects according to the semi-analytic Liu theory are expressed as (Liu and Alford, 1980), a_ m ¼ ha_ D i; X_ m ¼ hX_ G i þ hX_ D i e_ m ¼ h_eG i þ h_eD i; x_ m ¼ hx_ G i þ hx_ D i _ m ¼ hM _ G i þ hM _ Di i_m ¼ hi_G i þ hi_D i; M where h_xD i ¼

1 2p

Z

ð8Þ

2p

x_ D ðam ; em ; im ; xm ; MÞdM 0

x indicates any one of the six orbital elements. The upper dot denotes differentiation with respect to time, a is semimajor axis, e is eccentricity, i is inclination, X is ascending node, x is argument of perigee, and M is mean anomaly. The subscripts G and D represent the perturbing terms due to the Earth’s oblateness and the atmospheric drag, respectively. The symbol hxi represents the averaged value of x with respect to mean anomaly. Although the averaged equations can be evaluated numerically without simplification, some of the less important integrals are assumed negligible. Because a_ D and e_ D are the principal effects of the atmospheric drag for secular changes, the remaining _ D ) are ignored. Also, im is assumed to effects (i_D ; x_ D ; X_ D ; M

be constant because i_G is very small with a multiplier e or e2 . Detailed description can be found in Liu and Alford (1980). With these assumptions, the averaged equations of motion can be simplified and expressed as, a_ m ¼ ha_ D i; X_ m ¼ hX_ G i e_ m ¼ h_eG i þ h_eD i; x_ m ¼ hx_ G i _ m ¼ hM _ Gi i_m ¼ 0; M where

ð9Þ

" a BqV 1 þ e2 þ 2e cos f 1  e2 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 a3 ð1  e2 Þ3 5 xa cos i dM l

1 ha_ D i ¼  2p

1 h_eD i ¼  2p

Z

Z

2p

(

2p

BqV 0

)

r2 xa cosi e þ cosf  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½2ðe þ cos f Þ 2 lað1  e2 Þ

e sin f  dM 2

B is ballistic coefficient, q is air density, V is satellite velocity relative to ambient air, f is true anomaly, xa is atmospheric rotational speed, l is the product of gravitational constant and mass of the Earth, and r is the magnitude of position vector. The integrations, which give the averaged drag effects with respect to M from 0 to 2p, are calculated with respect to the true anomaly f by, dM ¼ ðr=aÞ2 ð1  e2 Þ

1=2

df

ð10Þ

B ¼ C D A=M s ð11Þ #  1=2 " 3=2 2 l ð1  e Þ x a V ¼ ð1 þ e2 þ 2e cos f Þ cos i 1 p 1 þ e2 þ 2ecos f n ð12Þ where C D is drag coefficient, A is cross-sectional area of the satellite, M s is total mass of the satellite, n is mean motion, and p = að1  e2 Þ. Detailed description can be found in Liu and Alford (1980). The aforementioned equations are solved using the fourth order Runge-Kutta numerical integration scheme as well as the Gauss-Legendre quadrature. The explicit _ G i can be found in Liu terms for hx_ G i; h_eG i; hX_ G i, and hM and Alford (1980). 2.3. Reentry survivability analysis In line with the representative reentry analysis codes such as the ORSAT, the SCARAB, and the Survivability Analysis Program for Atmospheric Reentry (SAPAR), the present module was made based on the well known Lees’ and Fay and Riddell’s formulae in the heat transfer

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rate calculation. The module contains six parts, namely, trajectory, atmosphere, aerodynamics, aerothermodynamics, thermal analysis, and ablation. Fig. 2 shows a diagram of the parts used in the reentry code. Detailed information can be found in our previous work (Park and Park, 2017) and so only a brief description is given here. Three-degrees-of-freedom (DOF) motion is calculated using the equations derived in an Earth-fixed reference frame assuming a rotating spherical Earth in the trajectory part. By using the fourth order Runge-Kutta numerical integration scheme, the equations can be solved. The atmosphere part provides the trajectory part with the atmospheric temperature, density, and pressure distribution as a function of altitude. The 1976 U.S. Standard Atmosphere model is used in line with the existing codes (Lips et al., 2005; Sim and Kim, 2011). The aerodynamics part calculates the average aerodynamic coefficient of a simpleshaped object such as sphere, box, flat plate, or cylinder for free molecular, transitional, and continuum flow regimes along with Knudsen number (Klett, 1964; Cropp, 1965). The aerothermodynamics part calculates the net heating rate to an object. It can be calculated from the hot wall convective heating rate plus the oxidation heating rate minus the re-radiation heating rate. The hot wall convective heating rate can be obtained as the product of the cold wall convective heating rate and the wall enthalpy ratio (Rochelle et al., 1997). The cold wall convective heating rate is calculated as the stagnation heating rate multiplied by a factor, F, to provide the average heating to the object. For continuum flow, the Detra, Kemp, and Riddell relation (Detra et al., 1957) is used to calculate the stagnation heating rate. For free molecular regimes, a relation consisted of free molecular thermal accommodation, density, and velocity is used (Bouslog et al., 1994). The relation between Stanton number and Knudsen number has been used for transitional flow regimes (Cropp, 1965). The oxidation heating rate is calculated using the simple formula from Cropp (1965). The Stephan-Boltzmann equation is used to calculate the heat loss caused by re-radiation.

Fig. 2. Logic schema of the reentry code (reproduced from Sim and Kim (2011)).

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In the thermal analysis part, surface temperature and inner temperature of an object are calculated. This part uses a nodal thermal math model. The differential equation for the 1-D thermal math model can be solved using a Forward Time Central Space (FTCS) finite difference scheme. The ablation part decides whether the outer layer of an object is eliminated or not. When the temperature is fixed at the melting temperature, the absorbed heat can reach the heat of ablation of the outer layer. Consequently, the layer is removed and the net heating rate is applied to the next layer. The changed size and ballistic coefficient of the object affect the other parts. The process continues until all the layers are melted or the object reaches the ground (Rochelle et al., 1997; Park and Park, 2017).

3. Results and discussion 3.1. Module validation The integrated system consists of the orbit prediction, the orbital lifetime analysis, and the reentry survivability analysis modules. To ensure the reliability of the system, each module was compared and validated with results of the existing codes. There are several free or commercial codes such as Orbit Extrapolation Toolkit (Orekit), Orbit Determination Toolbox (ODTBX), DRAMA, STELA, and Systems Tool Kit (STK) concerning the orbit and lifetime estimations (Maisonobe et al., 2010; Ko and Scheeres, 2014; Choi et al., 2017). In Korea, the STK has been mainly used by Korea Aerospace Research Institute (Choi et al., 2015). The STK from Analytical Graphics, Incorporated (AGI) provides high fidelity simulation and visualization of satellite behaviour including orbital dynamics, orbit collision avoidance, walker constellation, orbital lifetime, etc., and has been widely used over the last decade. Various researches have been conducted using the STK (Spangelo et al., 2013; Choi et al., 2015; Chamchalaem and Drake, 2016). Regarding the reentry survivability, the DAS by NASA and the DRAMA (SARA) by ESA are the most representative open codes. However, in the research institutes, the ORSAT and the SCARAB are mainly used as primary codes with high fidelity because the DAS and the DRAMA generate conservative survivability results using relatively simple approaches (Lips and Fritsche, 2005; Sim and Kim, 2011). Flight data for the reentry objects such as a Sandia fuel rod, Delta-II second stage fragments, and a ATV have been analyzed using the ORSAT or the SCARAB (Rochelle et al., 1997; Park and Park, 2017). In this work, the orbit prediction and the orbital lifetime analysis modules were validated and compared with results of the STK (Version 9). Although the algorithms of the STK are not open source, it is an accessible commercial code so the code to code comparison has been made using the same initial condition. For the reentry survivability

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analysis module, the results of the DRAMA (SARA) and the ORSAT were used. In literature, the survivability of a titanium tank loaded with frozen hydrazine reentering the Earth’s atmosphere was analyzed by the ORSAT (Kelley and Rochelle, 2012). Because the ORSAT is not open to the public and the use of the code is restricted by NASA, the results in literature were used for validating the present survivability analysis module. 3.1.1. Orbit prediction The SGP, SGP4, SDP4, SGP8, and SDP8 models were considered in the orbit prediction module. This set of models is often referred to as the SGP4 model because of its frequent use with the TLEs. Depends on whether the orbiting object is near-Earth or deep-space, the use of the SGP4 or the SDP4 model is made. The SGP8 and the SDP8 models also have the same atmospheric and gravitational models as the SGP4 and the SDP4 models, however the solution equations are considerably different. As mentioned, the TLEs must be used with one of the models to obtain the maximum prediction accuracy because they are obtained by removing periodic variation as suggested by literature (Hoots and Roehrich, 1980). To confirm the availability and the validity of the present module, the results were compared with that of the STK based on the same TLEs. Since the SGP4 model is

Table 3 Classical orbit elements for the Korean satellites at epoch time. Name

a (km)

e (–)

i (deg)

X (deg)

x (deg)

M (deg)

KOMPSAT-2 STSAT-2C KITSAT-1 KITSAT-3

7070.13 6872.05 7526.25 7097.93

0.000758 0.03161 0.019275 0.002568

98.03 80.11 90.27 98.44

209.57 281.36 283.24 11.39

306.81 335.03 92.04 81.72

53.20 23.46 23.16 278.47

only available in the STK, the comparison of the results of the SGP4 models was made. Korea Multi-purpose Satellite 2 (KOMPSAT-2), Science and Technology Satellite 2C (STSAT-2C), and Korea Institute of Technology Satellites 1 and 3 (KITSAT-1 and KITSAT-3) were used as test models to consider the various Korean satellites in the LEO. The orbit elements at epoch time and the TLEs for the models are shown in Tables 3 and 4, respectively. The TLEs can be obtained from Space-Track.org (https:// www.space-track.org/). As shown in Table 4, the first line represents satellite number, TLE’s epoch time, the first and the second derivatives of mean motion, and the drag term (B ), and the second line represents the classical Keplerian elements. Figs. 3–6 compare the position and velocity vectors of each test model predicted using the present approach with that of the STK. The graphs show that differences in the position and velocity vectors of both the present and the STK. The results were calculated for a year from each epoch time, and the time step was taken to be 1 min. Since the analytical method was used, there is no difference in the results according to calculation time step. In the figures, to better visualize the difference in the results between the two models, the figures were plotted 0–1 and 364–365 days, respectively. Looking at Fig. 3(a), while the upper figure represents the differences of the position vector components from epoch time to 1 day, the sub-figure shows them from 364 to 365 days. The differences are small between 0.005 and 0.005 km during the first day from epoch time, but they increase between 1.6 and 1.6 km by the end of the year. Fig. 3(b) shows the differences in velocity vector components, and the differences are relatively small when compared to that of position vector components. Figs. 4–6 also show similar results as in Fig. 3. Although the differences slightly increase over time, the results predicted by the present and the STK show good agreement in general.

Table 4 TLEs of the Korean satellites.

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Fig. 3. Comparison of the position and velocity vectors for KOMPSAT-2 (True Equator Mean Equinox (TEME) coordinate system).

The used algorithms and/or truncation errors are regarded as the main reason for the differences. 3.1.2. Orbital lifetime The orbital lifetime analysis module was compared and validated with results of the STK lifetime tool. The semianalytic theory by Liu and Alford is used for the nearEarth orbiting objects in the lifetime analysis module. The simplified averaged equations of motion, Eq. (9), were used and solved through the following procedures. Integration in the equations was performed using Gauss-Legendre formulas with a tenth order of quadrature evaluation. After the integration, the equations were solved using a

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Fig. 4. Comparison of the position and velocity vectors for STSAT-2C (True Equator Mean Equinox (TEME) coordinate system).

fourth order Runge-Kutta numerical scheme. For the atmospheric drag calculation, the 1976 U.S. Standard Atmosphere model which considers a steady-state atmosphere for moderate solar activity was considered in the present module. On the other hand, the STK lifetime tool has various density models and solar flux data which provide variation of the atmospheric density according to the solar activity. The STSAT-3 was used as test model and the TLEs are given in Table 5. To examine the effects of ballistic coefficient, atmospheric density model, and solar activity, in total 7 different cases were considered as initial condition one at a time. Drag area, mass, drag coefficient (C d ),

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Fig. 5. Comparison of the position and velocity vectors for KITSAT-1 (True Equator Mean Equinox (TEME) coordinate system).

Fig. 6. Comparison of the position and velocity vectors for KITSAT-3 (True Equator Mean Equinox (TEME) coordinate system).

ballistic coefficient (b), atmospheric density model, reflection coefficient (C r ), Area Exposed to Sun (AES), and solar flux data for the cases are shown in Table 6, and all the parameters are used as input conditions for the lifetime analysis. Because the present module does not have solar flux data, C r and AES are only used in the STK. The cases 1 to 3 were considered to investigate the effect of ballistic coefficient using the same density model. While the ballistic coefficient of the case 2 is two times lower than that of the case 1, the case 3 is two times higher than the case 1. The ballistic coefficients for the cases 4 to 7 are the same as that of the case 1. The cases 4 and 5 were considered

to examine the effect of atmospheric density model in which the Jacchia 1970 and 1971 models were used. The cases 6 and 7 were considered for the effect of solar activity using solar flux data named ‘SolFlx_Schatten’ or ‘SolFlx_0101’, where ‘SolFlx’ and ‘Schatten’ are the abbreviation of ‘Solar Flux’ and the last name of the writer, K. H. Schatten, respectively (Choi et al., 2015). ‘0101’ represents the month and the year which means the predicted solar flux from January 2001. The solar flux values are determined based on a particular deviation of the F10.7 cm radio emissions within a range of +2r prediction error (Choi et al., 2015).

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Table 5 TLEs of the STSAT-3.

Table 6 Summary of initial conditions. Case

Drag area (m2 )

Mass (kg)

C d (–)

b (kg/m2 )

Density model (–)

C r (–)

AES (m2 )

Solar flux data (–)

1 2 3 4 5 6 7

0.89 0.89 0.89 0.89 0.89 0.89 0.89

170 170 170 170 170 170 170

2.2 4.4 1.1 2.2 2.2 2.2 2.2

86.82 43.41 173.65 86.82 86.82 86.82 86.82

1976 Standard 1976 Standard 1976 Standard Jacchia 1970 Jacchia 1971 1976 Standard 1976 Standard

– – – – – 1.0 1.0

– – – – – 2.61 2.61

– – – – – SolFlx_Schatten SolFlx_0101

Fig. 7. Comparison of the semi-major axis for cases 1 to 3.

Fig. 8. Comparison of the inclination for cases 1 to 3.

Figs. 7–9 compare the semi-major axis, inclination, height of perigee and apogee predicted by the present approach and the STK lifetime tool. Fig. 7 represents the semi-major axis against time for the cases 1 to 3. Comparison was made from epoch time to the end of the orbital lifetime. Decay altitude at which the calculation ends was set at 90 km. The results show that the end of the orbital lifetime increases with an increase in the ballistic coefficient. Also, it can be seen that the profiles predicted by the STK and the present code are in good agreement. For further detailed comparison, the results predicted by the STK and the present method are presented in Table 7, where the percentage difference was calculated based on that of the STK. It can be noted that the lifetime differences between the STK and the present method are small.

Looking at Fig. 8, the inclination angle against time for the cases 1 to 3 can be seen. With the present results, the inclination angle remains constant regardless of the cases, because the simplified averaged equations of motion were used assuming the inclination is constant. Although the results of the STK show a slight variation of inclination, but the level of variation is small because the differences of inclination for the STK and the present method are between 0.3° and 0.2°. The perigee and apogee against time for the cases 1 to 3 can be seen in Fig. 9(a) and (b), respectively. From the definitions of the perigee and apogee, it can be seen that the height of apogee is relatively high compared to the perigee. Even though the present method and the STK use different approaches for the orbital lifetime analysis, both the methods show the results in Figs. 7–9 are still in good agreement.

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Fig. 10. Effects of atmospheric drag and solar activity on the semi-major axis and the orbital lifetime. Fig. 9. Height of perigee and apogee for cases 1 to 3.

Table 7 Lifetime comparisons for cases 1 to 3.

Case 1 Case 2 Case 3

STK (year)

Present (year)

Difference (%)

31.71 15.81 63.31

31.75 15.87 63.50

0.13 0.38 0.30

The effects of atmospheric density model and solar activity on the semi-major axis and the end of the orbital lifetime are shown in Fig. 10(a) and (b), respectively. The case 1 was compared with the cases 4 to 7 based on the same ballistic coefficient. The results of the cases 1, 6, and 7 are quite similar which indicates that the solar activity effect for the cases is small. For the cases 4 and 5, the

results of the end of the orbital lifetime were noticeably different. It can be noted that the uncertainties in the atmospheric densities and the ballistic coefficient have a larger impact on the prediction accuracies for low-altitude objects (Liu and Alford, 1980). However, although the effects of solar activity on the orbital lifetime analysis are small in the present cases, the solar activity effects can be significant for a long-term prediction, because the solar activity is a major factor for changes in the atmospheric density and consequently the orbital lifetime can vary along with a change in the solar activity (Kebschull et al., 2013; Choi et al., 2015). In addition to the solar activity, the external forces such as solar and lunar gravitational perturbations are assumed negligible in the present module. The assumptions may affect the lifetime as well as the reentry initial condition, especially for the deep-space objects. In this

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regard, the effects of reentry initial condition on the survivability are discussed in Section 3.3 with the parametric analysis including true anomaly. 3.1.3. Reentry survivability For the validation of the reentry survivability analysis module, the results of the DRAMA and the ORSAT were used. First, the present module was validated by comparing the analysis results of sphere to the results of the DRAMA based on the same condition. In total, 6 sphere cases having different sizes and materials were considered (see Table 8). Since the DRAMA only uses a lumped thermal mass model, the fully solid spheres, not the hollow spheres, were examined. In addition, the oxidation heating was not considered and the temperature independent material properties were used along with the DRAMA. The material properties can be found in Park and Park (2017). The DRAMA assumes that the reentry object is decomposed into its individual elements at an altitude of 78 km. Therefore, in the present module, it is considered that the separation process occurs at 78 km. Semi-major axis, eccentricity, inclination, RAAN, Argument of perigee, and true anomaly were 6470 km, 0.001, 10°, 7.3°, 12°, and 200°, respectively, for the initial condition. Because the detailed results such as heating rates are not given from the DRAMA, the results of survivability and downrange were compared. Fig. 11 shows the variation of altitude against downrange. Fig. 11(a) and (b) represent the results of the DRAMA and the present module, respectively. The results for both the codes show the same survivability, which 4 out of 6 cases survived. Furthermore, the cases 1 and 4 demise at 45.6 and 64.3 km for the DRAMA, and demise at 45 and 63.6 km for the present module. The demise altitudes are quite similar (within 1.4%). However, when the downranges are compared, there is some discrepancy for the survived cases. The different Earth models including the Earth’s gravity are regarded as the reason for this cause. NASA analyzed the survivability of a titanium tank loaded with frozen hydrazine that reenters the Earth’s atmosphere using the ORSAT code (Kelley and Rochelle, 2012). The titanium tank results have been used by various researchers including code validation (Sim et al., 2015). The present authors also used the titanium results for the survivability analysis module validation. The titanium tank with frozen hydrazine (N2H4) was modeled as a spinning

Fig. 11. Variation of altitude against downrange for six cases.

shell-sphere, consisted of five titanium and one hydrazine nodes. The tank had diameter of 1.04 meters with wall thickness of 0.00356 meters. It had 454 kg of N2H4. Initial temperature of the tank and the N2H4 was assumed to be 214 K. It was assumed that initial altitude was 78 km, the

Table 8 Physical properties for sphere. Case

Material

Outer Radius (m)

Inner Radius (m)

Mass (kg)

1 2 3 4 5 6

Graphite epoxy 2 Titanium Aluminum Graphite epoxy 2 Titanium Aluminum

0.50 0.50 0.50 0.25 0.25 0.25

0 0 0 0 0 0

811.84 2303.83 1413.72 101.48 287.98 176.71

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S.-H. Park et al. / Advances in Space Research xxx (2018) xxx–xxx Table 9 Hydrazine and Titanium properties. Property Specific heat capacity, C p (J/kg-K) Thermal conductivity, k (W/m-K) Density, q (kg/m3) Emissivity, e (–) Heat of fusion, hf (J/kg) Oxide heat of formation, DH OX (J/kg-O2) Melting temperature, T m (K)

Hydrazine (N2H4)

Titanium (6 Al-4 V)

1559 2.4 1025 0.5 395025 0 275

(Kelley and Rochelle, 2012) (Kelley and Rochelle, 2012) 4437 (Kelley and Rochelle, 2012) 393559 32481250 1943

relative velocity and flight path angle were 7.58 km/s and 0.2°, respectively. The properties of the titanium (6 Al4 V) and the frozen hydrazine (N2H4) are given in Table 9. Figs. 12 and 13 represent surface temperature and surface heating rates of the tank over time, respectively. Overall, the comparison for each temperature and heating rate shows similar results. It is to be noted that the present results show that 2 out of 6 nodes survived including one hydrazine node in which the outcomes are the same as that of the ORSAT. The present results show good agreement with that of the ORSAT in terms of survivability. 3.2. Case study - analysis of STSAT-3

Fig. 12. Variation of surface temperature against time for hydrazine propellant tank.

3.2.1. Orbital lifetime One practical test model was considered for the orbital lifetime and the reentry survivability analysis. The model was satellite STSAT-3, which is a small near-Earth orbiting satellite designed for a long term mission by Korea Aerospace Research Institute (KARI). The satellite was launched in November 2013 in Russia and it represents the third experimental mini-satellite of the STSAT series, within a stream of long term national space development plan by the Ministry Of Science and Technology (MOST) of Korea. The orbital lifetime of the STSAT-3 is computed using the TLEs, and the TLEs are shown in Table 5. The effects of ballistic coefficient, atmospheric density model, and solar activity on the lifetime were discussed in the previous Section 3.1. In the present module, the 1976 U.S. Standard Atmosphere model was used and the solar flux data was not included. To calculate the orbital lifetime, the following characteristics of the STSAT-3 were considered based on

Table 10 Final orbital elements of the STSAT-3.

Fig. 13. Variation of surface heating rates against time for hydrazine propellant tank.

Semi-major axis, a (km) Eccentricity, e (–) Inclination, i (°) Argument of perigee, x (°) RAAN, X (°) True anomaly, h (°)

6500 0.00054 97.66 240.36 215.27 175.87

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the available information online. Drag area and mass were 0.89 m2 and 170 kg, respectively. Drag coefficient was assumed to be 2.2 because the used drag coefficients for Korean satellites (KOMPSAT-1 and KOMPSAT-2) were between 2.0 and 2.4 in literature (Choi et al., 2015). In the calculation, the STSAT-3 lifetime was estimated to be 31.75 years (1001376000 s) from epoch time, and orbit count was 174844 orbits. Table 10 shows the final orbital elements. 3.2.2. Reentry survivability The STSAT-3 includes Multi-purpose Infrared Imaging System (MIRIS), Command and Data Handling Subsystem (CDS), Attitude Control Subsystem (ACS), and Compact Imaging Spectrometer (COMIS), in which subcomponents are mostly in a form of boxes or spheres. Because information regarding the number of parts used for the STSAT-3 is not open to the public, to describe the parts as much as possible, it was assumed that the satellite is composed of 12 parts (6 boxes and 6 spheres) herein. To analyze the reentry survivability according to various ballistic coefficients and also to better view the downrange spectrum, the parts having somewhat arbitrary sizes and masses were considered. The spheres were modeled as a spherical shell. The boxes were modeled to have the same width and height. The 12 parts are listed in Table 11. The materials considered were graphite epoxy (GrEp) 1,2, aluminum (Al), and titanium (Ti). Their properties can be found in Park and Park (2017). Initial condition for the reentry was obtained from the results of the orbital lifetime analysis through the coordinate transformation. The condition is included in Table 12. Fig. 14 presents the variation of stagnation-point heating rates against time for the 12 parts. The predicted heating rates for the boxes and the spheres can be seen in Fig. 14(a) and (b), respectively. Looking at Fig. 14(a), the part 2 shows the highest peak heating rate when compared with the other boxes and spheres. When the parts 4 and 5 demise, they reach a peak heating rate of 1,317,700 W/m2 and 181,730 W/m2, respectively. Looking at Fig. 14(b), a

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Table 12 Initial condition for the reentry analysis. Altitude, H (km) Velocity, V (km/s) Flight path angle, b (°) Azimuth, A (°) Latitude, d (°) Longitude, a (°)

126 7.83 0.012 103.7 55.47 23.94

similar observation is made for the spheres. The part 7 represents the highest peak heating rate compared with that of the other spheres. The parts 8, 9, and 12 reach a peak heating rate of 1,080,000 W/m2, 486,050 W/m2, and 95,237 W/m2, respectively, when they demise. Fig. 15 shows the variation of altitude against downrange. The downrange is different for each part due to the difference in the ballistic coefficient used. The results show that 5 out of 12 parts demised. The reason that the parts 4, 5, 8, 9, and 12 demise is because those are made of aluminum or graphite epoxy 2 that has lower melting temperature as well as heat of fusion than the other materials. Fig. 16 presents the variation of velocity against downrange. Again, it can be seen that 5 out of 12 parts demised similarly to the results in Fig. 15. Initially, the velocity is maintained due to low air density, but the reason for a rapid drop in the velocity after some time is due to an increase in the drag induced by high air density (Park and Park, 2017). 3.3. Effect of true anomaly 3.3.1. Circular orbit Throughout the orbital lifetime analysis, all the six orbital elements (semi-major axis, eccentricity, inclination, argument of perigee, RAAN, and true anomaly) were obtained. From the results, the circular orbit of the STSAT-3 was determined because the obtained eccentricity (e) was 0.00054 which is a small value. The true anomaly (h) defines the orbiting object position along the orbit at

Table 11 Part details of the STSAT-3. Part

Shape

Material

Outer Width (m)

Inner Width (m)

Mass (kg)

Length (m)

1 2 3 4 5 6

Box Box Box Box Box Box

Al Ti GrEp 1 GrEp 2 Al Ti

0.250 0.250 0.250 0.250 0.500 0.500

0.134 0.136 0.134 0.134 0.498 0.499

60.075 97.000 34.930 34.490 12.685 12.685

0.500 0.500 0.500 0.500 2.500 2.500

Outer Radius (m)

Inner Radius (m)

0.125 0.125 0.250 0.250 0.500 0.500

0.075 0.075 0.243 0.243 0.498 0.498

7 8 9 10 11 12

Sphere Sphere Sphere Sphere Sphere Sphere

GrEp GrEp Al Ti GrEp GrEp

1 2

1 2

10.070 9.945 14.432 23.519 12.269 12.117

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Fig. 15. Variation of altitude against downrange.

Fig. 14. Variation of the stagnation-point heating rates against time. Fig. 16. Variation of velocity against downrange.

epoch. The remaining five orbital elements determine the shape and the size of the orbit as well as the orientation of the orbital plane. Contrary to the present module, the true anomaly calculation is not included in the STK lifetime tool. Therefore, the general characteristics of the orbit can be determined, but the actual position of the object in the orbit is not attainable due to the unknown true anomaly. Although the true anomaly may be difficult to predict accurately because it is 2p periodic in the fast variable, it is the vital element to estimate the reentry trajectory. To examine the effect of true anomaly in a circular orbit on the reentry survivability, four values of true anomaly: 0°, 90°, 180°, and 270° were used. Figs. 17 and 18 show the demise altitudes and impact masses of the 12 parts with the variation of true anomaly. Also, the results of h ¼ 0 were compared with that of 90°, 180°, and 270°. In regard

to Fig. 17(a), the 7 out of 12 parts survived and reached the ground with different true anomalies. Although the parts 4, 8, and 9 show a slightly large difference of the demise altitude when compared with the other parts, the number of the survived parts is the same for the anomalies considered. Fig. 17(b) presents the demise altitudes for h ¼ 0 compared with the 90°, 180°, and 270° cases. The determination coefficient (R2) is a statistic parameter that explains how close the points are to the trend line. If all points are along the red line (y = x, R2 = 1), the results agree perfectly (Park and Park, 2017). It can be seen that the values of R2 and the slopes are close to unity, which indicates that the results of the h ¼ 0 show good agreement with that of the 90°, 180°, and 270°.

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Fig. 17. Variation of true anomaly on demise altitudes in a circular orbit.

Fig. 18. Variation of true anomaly on impact masses in a circular orbit.

Considering Fig. 18(a), it also shows that the 7 out of 12 parts survived similarly to the results in Fig. 17(a). Overall, the predicted impact masses show good agreement except the parts 1 and 2. The predicted impact masses for the parts 1 and 2 show a difference, but the prediction of the survivability is the same for the four different true anomalies. Fig. 18(b) presents the impact masses for h ¼ 0 compared with the 90°, 180°, and 270° cases. The results show good agreement because the values of R2 and the slopes are close to unity. Even though the results of the demise altitude and impact mass for the four true anomalies do not represent a perfect match, Figs. 17 and 18 show the same survivability for the four different true anomalies. To conduct the sensitivity analysis of the true anomaly for the reentry survivability in a circular orbit, a Monte Carlo simulation was performed. 500 simulations for each part were conducted according to variation of true anomaly between 0 and 2p. Fig. 19(a) and (b) show the demise altitudes and the impact masses for true anomaly variations, respectively. As shown in Figs. 17 and 18, the results

show that 7 out of 12 parts survived with 500 different true anomalies. Although there are discrepancies in both the demise altitude and the impact mass, it indicates that the survivability does not change according to different true anomalies for the present parts considered. For orbiting object with small eccentricity and low altitude, the orbit shape of the orbiting object becomes circular rather than elliptical by the end of its lifetime, because the object loses its energy due to factors such as the atmospheric drag. For this reason, the eccentricity of the STSAT-3 in the lifetime analysis is very close to zero meaning ‘circular orbit’. Altitude, velocity, and flight path angle of the orbiting object when it reenters are almost the same regardless of the true anomaly due to the fact that the radius and the angular velocity of the orbiting object on the circular orbit are constant. Hence, there was no significant difference in survivability depending on the true anomaly and the reason will be discussed in Section 3.3.3. Consequently, even if the true anomaly may not be predicted, no strong effect on the survivability is expected as long as the circular orbit is concerned.

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Fig. 20. Variation of true anomaly on demise altitudes in an elliptic orbit. Fig. 19. Demise altitudes and impact masses for variation of true anomaly from Monte Carlo simulation.

3.3.2. Elliptic orbit When the eccentricity is in between 0 and 1, the orbit shape of an orbiting object is elliptical. Hohmann transfer, Molniya, and tundrat orbits are the examples of the elliptic orbit. Using the orbital elements of the STSAT-3 (Section 3.2) assuming the reentry orbit is not circular but elliptic, the effect of true anomaly on the reentry survivability is investigated. Again, four true anomalies (h ¼ 0 , 90°, 180°, and 270°) were considered. Eccentricity and semi-major axis were assumed to be 0.1 and 7100 km, respectively. Figs. 20 and 21 show the demise altitudes and impact masses of the 12 parts with the variation of true anomaly. Concerning Fig. 20(a), the results show totally different survivability depending on the true anomaly. While the 4 out of 12 parts demised in the atmosphere with h ¼ 180 and 270°, the 6 and 5 parts demised with 0° and 90°, respectively. For the demised parts, each part shows a large difference of the demise altitude. The demise altitudes for h ¼ 0 are compared with that of 90°, 180°, and 270° in

Fig. 20(b). The results show the values of R2 and the slopes are far from unity. It can be noted that the demise altitudes are different in accordance with the true anomaly. Fig. 21(a) shows the impact masses of the 12 parts in the elliptic orbit. While the 8 out of 12 parts survived with h ¼ 180 and 270°, the 6 and 7 parts survived with 0° and 90°, respectively. Even though the overall results of the impact masses are similar, there is a large difference in the survivability. The impact masses for h ¼ 0 are compared with that of 90°, 180°, and 270° in Fig. 21(b). Considering the R2 and the slopes, the results are clearly different. The number of the survived parts is also different. Contrary to the survivability results of the circular orbit, Figs. 20 and 21 show a large difference in the survivability for the elliptic orbit. It can be noted that there is a strong effect on the survivability depends on the true anomaly used because when the orbiting object reenters from the elliptic orbit, the initial condition for altitude, velocity, and flight path angle varies according to the true anomaly. For the four true anomalies (h ¼ 0 , 90°, 180°, and 270°) considered, the ranges of initial altitude, velocity, and flight

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to various initial velocities and flight path angles. Since the heating rate only slightly increases in the free molecular regime where the Knudsen number is much greater than one, the altitude was not considered. Fig. 22 shows the stagnation-point heating rates over time for various initial velocities and flight path angles. To compare the results in detail, time in abscissa is shifted for the time of maximum heating rate to be located on zero. Looking at Fig. 22(a), as the velocity increases from 7.63 to 7.83 km/s, the maximum stagnation-point heating rate decreases. While the maximum heating rate is about 1,480,000 W/m2 when the initial velocity is 7.63 km/s, the maximum heating rate is about 1,270,000 W/m2 when the initial velocity is 7.83 km/s. However except for the maximum heating rate regime, the heating rate increases as

Fig. 21. Variation of true anomaly on impact masses in an elliptic orbit.

path angle were between 77 and 1510 km, 6.74 and 8.24 km/s, and 5.7° and 5.7°, respectively. The true anomaly effect on both the reentry trajectory and survivability is regarded to be important as long as the elliptic orbit is considered. 3.3.3. Parametric analysis In the reentry survivability analysis, the heat flux is directly related to the survivability of reentry objects (Detra et al., 1957; Rochelle et al., 1997; Sim and Kim, 2011). As the heat flux increases, the survivability decreases and vice versa (Park and Park, 2017). According to the Detra, Kemp, and Riddell formula, the stagnation-point heating rate is calculated from the atmospheric density and velocity of the reentry object (Detra et al., 1957). Therefore, altitude (H), velocity (V), and flight path angle (b) among the six reentry input parameters are considered to be the main factors of survivability. The parametric analysis was performed to examine the effects of initial velocity and flight path angle on the survivability. Based on the reentry initial condition in Table 12, the heating rates of titanium sphere (part 10) were compared according

Fig. 22. Variation of the stagnation-point heating rates on initial velocity and flight path angle.

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the initial velocity increases. So, the whole heat flux integrated during the falling time becomes larger when initial velocity increases, leading to low survivability of the reentry objects. The stagnation-point heating rates for a range of initial flight path angle from 0° to 2° are presented in Fig. 22(b). As the initial flight path angle decreases, the maximum stagnation-point heating rate increases. The flight path angles of 0° and 2° correspond to the maximum heating rates of 1,270,000 W/m2 and 1,680,000 W/m2, respectively. As discussed in Fig. 22(a), as the initial flight path angle increases, the integrated heat flux increases, resulting in the low survivability. Fig. 22 indicates that among the six initial parameters, the velocity and the flight path angle are particularly more important for the reentry survivability. In the previous sections, there was no significant effect on the survivability on different true anomalies for the circular orbit in which eccentricity is very close to zero, and the initial altitudes, velocities, and flight path angles among the six reentry parameters (H ; V ; b; A; d, and a) were similar. For the elliptic orbit, however, a large difference in the survivability with all the different reentry parameters was found. It can be noted that the effects of initial velocity and flight path angle on the survivability of reentry objects are significant, implying the importance of considering whether the reentry orbit is circular or not. 4. Conclusions Integrated system for the orbit, the orbital lifetime, and the reentry survivability estimation of the orbiting objects was developed. To ensure the reliability of the system, each module was compared and validated with the results of the existing codes based on the same initial conditions. Good agreement was found. To the authors’ understanding, for the first time, this work shows an overall procedure for the method of the integrated system within a single manuscript. The STSAT-3 was considered as the test model. The orbital lifetime was analyzed using the semi-analytic theory with the TLEs of the STSAT-3. The orbital elements at the end of the orbital lifetime were used as the initial condition for the reentry analysis. In particular, four values of true anomaly: 0°, 90°, 180°, and 270° were considered as the reentry initial condition to investigate the true anomaly effect, and the sensitivity analysis of the true anomaly was conducted using the Monte Carlo method. The results have shown that the number of the survived parts of the STSAT-3 is the same regardless of the different true anomaly. It has been demonstrated that there is no strong effect on the survivability for different true anomalies considered because the near-Earth satellites with small eccentricity reach a circular orbit at the end of it’s lifetime due to its energy loss. On the other aspect, the effect of true anomaly on the survivability was conducted in the elliptic orbit. Contrary to the circular orbit, the results showed a large difference in the survivability according to the different true anomalies. It was found that the true anomaly effect is

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