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Reducing the collision risk for high inclination constellations in the geosynchronous region
Advances in Space Research 34 (2004) 1193–1197 www.elsevier.com/locate/asr
Reducing the collision risk for high inclination constellations in the geosynchronous region I. Wytrzyszczak *, E. Wnuk, J. Kaczmarek Astronomical Observatory of A. Mickiewicz University, Słoneczna 36, 60-286 Poznan, Poland Received 30 November 2002; received in revised form 5 February 2003; accepted 10 February 2003
Abstract The goal of the paper was to determine orbital elements of an inclined, geosynchronous satellite so that the number of approaches to the real objects from the geostationary belt during its whole mission be the smallest. Two numerical simulations were performed on a time span of 3 years. They proved that less hazardous orbits are those with initial perigees lying in the vicinity of the equator (x 0 or x 180), and that orbits with low eccentricities are responsible for a higher number of close approaches. 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Space debris; Collision risk; Geosynchronous region; High inclination constellations
1. Introduction The geostationary ring is of special importance for telecommunication, space science and Earth observations. More than 800 satellites and rocket upper stages have been inserted into geostationary orbits. Nowadays, several individual missions and constellations are planned to be operational on geosynchronous orbits (GSO). They hold the same orbital period as a GEO satellite, but – having inclinations greater than 15 – they do not belong to the GEO belt. As an example we can mention the FAME (Full-sky Astrometric Mapping Explorer) mission, planned to be launched at the GSO with the inclination 28.7. For simple geometric reasons it is obvious that an object on a circular GSO orbit will cross the crowded geostationary belt two times per day, posing a risk of collisions to both operational and inactive GEO objects. Thus, the orbital elements of geosynchronous missions have to be planned with a great precaution. In order to study the collision risk, we have generated two sets of GSO orbits. Orbital elements of the first group served to determine a range of optimal arguments *
Corresponding author. Tel.: +48-61-829-2774; fax: +48-61-8292772. E-mail address: [email protected] (I. Wytrzyszczak).
of perigee, whereas the second group was formed to find which eccentricities and inclinations are more ÔsafeÕ. The paper is organized as follows: the first section describes the method of calculations with a detailed characteristic of generated GSO orbits. Sections 3 and 4 present the results of simulations aiming to determine a low-risk range of orbital arguments of perigee as well as eccentricities and inclinations. In the last section the analysis of two possible sources of errors is carried out.
2. Method of calculation The Bulirsch-Stoer integrator was applied to the equations of motion in rectangular coordinates. The perturbing forces considered in our model included J2 , J3 and J4 zonal harmonics, tesseral harmonics up to degree and order 4, lunisolar perturbations and the solar radiation pressure (neglecting both the radiation reflected by the Earth and a shadow function). Initial conditions for the real satellites were taken from the M. McCants web page (http://users2.ev1.net/~mmccants/). The catalogue of July 2002 provided the orbits of 795 satellites. The DISCOS database served as the source of areato-mass ratios S=m (Flury and Klinkrad, 2002), but it
0273-1177/$30 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2003.02.037
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I. Wytrzyszczak et al. / Advances in Space Research 34 (2004) 1193–1197
was quite difficult to recover the appropriate reflectivity coefficients Cr , especially for old satellites. In these circumstances we decided to attribute to each satellite the same value of Cr ¼ 1:3. The initial epoch T0 of the integration was 21.0 July 2002. Geostationary satellites with initial elements given for a different epoch were integrated before the simulation start in order to obtain a coherent set of osculating elements for the selected T0 . We integrated two groups of fictitious geosynchronous object in order to test the risk of their collision with real
geostationary objects. Both groups consisted of the objects with the same orbital semi-major axes a ¼ 42164 km, and initial mean anomalies M ¼ 0. • The first group contained 5184 objects. They shared the same initial inclination i ¼ 28:7 and eccentricity e ¼ 0:007. Their arguments of perigee x and right ascensions of ascending nodes X covered the range of 0 to 360 with the step of 5. All objects had the areato-mass ratio S=m ¼ 0:11 m2 /kg and the reflectivity coefficient Cr ¼ 1:3.
Fig. 1. The number of approaches closer than 100 km (top) and than 50 km (bottom) between GEO objects and a specific GSO orbit as a function of its argument of perigee and of the right ascension of the ascending node.
I. Wytrzyszczak et al. / Advances in Space Research 34 (2004) 1193–1197
• For the second group of objects the common elements were x ¼ 0 and X ¼ 44, thus initially all fictitious geosynchronous satellites crossed the equator at the stable geostationary point with longitude 105 W. Orbital inclinations of the second group varied in the range of 28 6 i 6 59 every 1 ; eccentricities were taken from 0:0001 6 e 6 0:01 with the step of 0.0005. The number of the objects was 693.
3. Simulation 1: the x diagram The equations of motion for the first sample (with a common eccentricity and inclination) were integrated on the time span of 3 years. Orbits of real geostationary satellites, as well as those of geosynchronous test objects, evolved under the action of the considered perturbing forces. The Cartesian coordinates of all objects were stored every 1 h of the simulated motion. After finishing the numerical integration, the distances between each geosynchronous object and all
1195
geostationary satellites were calculated from the stored data and a number of approaches closer than 100 km and closer than 50 km was counted. The exemplary results are presented in Fig. 1. They show the number of approaches closer than 100 km (top) and closer than 50 km (bottom), between a fictitious geosynchronous objects and all catalogued satellites from the geostationary belt as a function of initial arguments of perigee and of the right ascensions of the ascending nodes. Different types of contour lines (dashed, continuous), in different shades of gray refer to a given number of registered approaches. As a matter of fact, the results confirm that on a relatively short time scale, a purely Keplerian reasoning could be sufficient. Having the semi-major axis a equal to the radius of the geostationary ring A, the inclined, eccentric orbits with the lines of nodes lying in the equatorial plane x ¼ 0 or x ¼ 180 will cross this plane at rmin ¼ að1 eÞ < A or rmax ¼ að1 þ eÞ > A. Let f0 ¼ arccosðeÞ defining the true anomaly of the point having r ¼ A. Then only the orbits with initial x ¼ f0 or x ¼ f0 þ 180 will exactly cross the geostationary
Fig. 2. The number of approaches closer than 250 km (top) and than 100 km (bottom) as a function of the eccentricity and the inclination of a geosynchronous orbit.
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I. Wytrzyszczak et al. / Advances in Space Research 34 (2004) 1193–1197
ring. For small eccentricities the values are close to x 90 and x 270.
4. Simulation 2: eccentricity–inclination plot The integration of the equations of motion for the second group (with common lines of apsides) was performed on the time span of 1 year, starting from T0 . In this case, the fictitious geosynchronous objects moved
on simple Keplerian orbits in order to simulate them as being active, maneuverable satellites. The real satellites, as previously, were subjected to all perturbing forces. Fig. 2 shows the number of approaches closer than 250 km (top) and 100 km (bottom) as a function of eccentricity and inclination of fictitious objects. For the reasons discussed in the previous section, only low eccentricities and, mostly, low inclinations can be responsible for a higher number of close approaches.
Fig. 3. The error in distance between the geosynchronous object and the 02029D geostationary satellite caused by an incorrect value of the geostationary satelliteÕs reflectivity coefficient.
Fig. 4. The error in distance between the same satellites as in Fig. 3, but resulting from an inaccurate value of the geostationary satelliteÕs area-tomass ratio.
I. Wytrzyszczak et al. / Advances in Space Research 34 (2004) 1193–1197
5. Error estimation Two possible sources of errors were estimated: incorrect values of reflectivity coefficients and incorrect values of S=m. Figs. 3 and 4 present the distance between the geostationary satellite 02029D and a geosynchronous object with e ¼ 0:007, i ¼ 28:7, x ¼ 0, X ¼ 45, after approximately one year of the simulated motion. In Fig. 3, the distance is plotted for three values of the geostationary satelliteÕs reflectivity coefficient: Cr ¼ 1, Cr ¼ 1:3 and Cr ¼ 1:6. The error in the distance of the two satellites after one year depends almost linearly on the adopted value of Cr . The difference DCr ¼ 0:1 generates in this case an error of about 3 km. Similar results, of the order of several km, were obtained for other satellites. Fig. 4 shows errors in satellitesÕ distance that result from neglecting the S=m variations caused by an aspherical shape of the object and its variable orientation with respect to the Sun. The same satellite 02029D was taken as an example. Depending on the orientation of the object, its S=m varies from 0.0044 to 0.0105 m2 /kg, with the mean value S=m ¼ 0:0089 m2 /kg. The error imposed by the variability of S=m amounts to about 20 km/yr.
6. Conclusions We were looking for optimal orbital elements set of a geosynchronous satellite in order to avoid close approaches to objects within the geostationary belt defined as a segment of a spherical shell A DR 6 r 6 A þ DR (DR ¼ 75 km) and delimited by 15 in latitude. Potentially most harmful objects move on orbits with perigees close to x 90 or 270. Every geosynchronous orbit of that type immediately crosses the GEO ring at the radial distance p ¼ Aðl e2 Þ if its eccentricity is less pffiffiffiffiffiffiffiffiffiffiffiffi than DR=A 0.042. The range of hazardous initial eccentricities shrinks as the direction of the line of apsides approaches that of the line of node. In the limit case, when x ¼ 0 or x ¼
1197
180, only orbits with e < DR=A 0.0018 can enter the GEO belt. The number of approaches varies a little in the range of considered GSO inclinations ð28 6 i 6 59Þ, slightly decreasing for larger inclinations. This conclusion apparently contradicts the previous results (Kessler, 1981; € Opik, 1951) that indicated the proportionality of the collision probability to sinði=2Þ= sin i. However, the classical collision rate estimates were developed under the assumption that the distribution of ‘‘target’’ objects is homogeneous in the volume of space at a given latitude. In the case of the GEO belt the number of collisions depends on individual contributions from all small layers between 15. Due to ‘‘librational’’ character of the relationship between orbital inclinations and longitudes of nodes (see for instance Wytrzyszczak, 2004) the density of nonactive geostationary objects falls down with latitude and is not homogeneous over geographical longitudes.
Acknowledgements The area-to-mass ratios data used in this paper were available thanks to the kind help of Prof. W. Flury and Dr. H. Klinkrad. The work presented in this paper was performed as a part of the KBN Grants number 8T12E 028 20 and 5T12D 026 23.
References Flury, W., Klinkrad, H., 2002 (private communication). Kessler, D.J. Derivation of the collision probability between orbiting objects; the lifetime of JupiterÕs outer moons. Icarus 48, 39–48, 1981. € Opik, E.J. Collision probability with the planets and distribution of planetary matter. Proc. Roy. Irish Acad. 54, 161–199, 1951. Wytrzyszczak, I. Testing the safety of decommissioned spacecraft above GEO. Adv. Space Res., this issue, 2004 (doi:10.1016/j.asr. 2003.02.036).
Reducing the collision risk for high inclination constellations in the geosynchronous region I. Wytrzyszczak *, E. Wnuk, J. Kaczmarek Astronomical Observatory of A. Mickiewicz University, Słoneczna 36, 60-286 Poznan, Poland Received 30 November 2002; received in revised form 5 February 2003; accepted 10 February 2003
Abstract The goal of the paper was to determine orbital elements of an inclined, geosynchronous satellite so that the number of approaches to the real objects from the geostationary belt during its whole mission be the smallest. Two numerical simulations were performed on a time span of 3 years. They proved that less hazardous orbits are those with initial perigees lying in the vicinity of the equator (x 0 or x 180), and that orbits with low eccentricities are responsible for a higher number of close approaches. 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Space debris; Collision risk; Geosynchronous region; High inclination constellations
1. Introduction The geostationary ring is of special importance for telecommunication, space science and Earth observations. More than 800 satellites and rocket upper stages have been inserted into geostationary orbits. Nowadays, several individual missions and constellations are planned to be operational on geosynchronous orbits (GSO). They hold the same orbital period as a GEO satellite, but – having inclinations greater than 15 – they do not belong to the GEO belt. As an example we can mention the FAME (Full-sky Astrometric Mapping Explorer) mission, planned to be launched at the GSO with the inclination 28.7. For simple geometric reasons it is obvious that an object on a circular GSO orbit will cross the crowded geostationary belt two times per day, posing a risk of collisions to both operational and inactive GEO objects. Thus, the orbital elements of geosynchronous missions have to be planned with a great precaution. In order to study the collision risk, we have generated two sets of GSO orbits. Orbital elements of the first group served to determine a range of optimal arguments *
Corresponding author. Tel.: +48-61-829-2774; fax: +48-61-8292772. E-mail address: [email protected] (I. Wytrzyszczak).
of perigee, whereas the second group was formed to find which eccentricities and inclinations are more ÔsafeÕ. The paper is organized as follows: the first section describes the method of calculations with a detailed characteristic of generated GSO orbits. Sections 3 and 4 present the results of simulations aiming to determine a low-risk range of orbital arguments of perigee as well as eccentricities and inclinations. In the last section the analysis of two possible sources of errors is carried out.
2. Method of calculation The Bulirsch-Stoer integrator was applied to the equations of motion in rectangular coordinates. The perturbing forces considered in our model included J2 , J3 and J4 zonal harmonics, tesseral harmonics up to degree and order 4, lunisolar perturbations and the solar radiation pressure (neglecting both the radiation reflected by the Earth and a shadow function). Initial conditions for the real satellites were taken from the M. McCants web page (http://users2.ev1.net/~mmccants/). The catalogue of July 2002 provided the orbits of 795 satellites. The DISCOS database served as the source of areato-mass ratios S=m (Flury and Klinkrad, 2002), but it
0273-1177/$30 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2003.02.037
1194
I. Wytrzyszczak et al. / Advances in Space Research 34 (2004) 1193–1197
was quite difficult to recover the appropriate reflectivity coefficients Cr , especially for old satellites. In these circumstances we decided to attribute to each satellite the same value of Cr ¼ 1:3. The initial epoch T0 of the integration was 21.0 July 2002. Geostationary satellites with initial elements given for a different epoch were integrated before the simulation start in order to obtain a coherent set of osculating elements for the selected T0 . We integrated two groups of fictitious geosynchronous object in order to test the risk of their collision with real
geostationary objects. Both groups consisted of the objects with the same orbital semi-major axes a ¼ 42164 km, and initial mean anomalies M ¼ 0. • The first group contained 5184 objects. They shared the same initial inclination i ¼ 28:7 and eccentricity e ¼ 0:007. Their arguments of perigee x and right ascensions of ascending nodes X covered the range of 0 to 360 with the step of 5. All objects had the areato-mass ratio S=m ¼ 0:11 m2 /kg and the reflectivity coefficient Cr ¼ 1:3.
Fig. 1. The number of approaches closer than 100 km (top) and than 50 km (bottom) between GEO objects and a specific GSO orbit as a function of its argument of perigee and of the right ascension of the ascending node.
I. Wytrzyszczak et al. / Advances in Space Research 34 (2004) 1193–1197
• For the second group of objects the common elements were x ¼ 0 and X ¼ 44, thus initially all fictitious geosynchronous satellites crossed the equator at the stable geostationary point with longitude 105 W. Orbital inclinations of the second group varied in the range of 28 6 i 6 59 every 1 ; eccentricities were taken from 0:0001 6 e 6 0:01 with the step of 0.0005. The number of the objects was 693.
3. Simulation 1: the x diagram The equations of motion for the first sample (with a common eccentricity and inclination) were integrated on the time span of 3 years. Orbits of real geostationary satellites, as well as those of geosynchronous test objects, evolved under the action of the considered perturbing forces. The Cartesian coordinates of all objects were stored every 1 h of the simulated motion. After finishing the numerical integration, the distances between each geosynchronous object and all
1195
geostationary satellites were calculated from the stored data and a number of approaches closer than 100 km and closer than 50 km was counted. The exemplary results are presented in Fig. 1. They show the number of approaches closer than 100 km (top) and closer than 50 km (bottom), between a fictitious geosynchronous objects and all catalogued satellites from the geostationary belt as a function of initial arguments of perigee and of the right ascensions of the ascending nodes. Different types of contour lines (dashed, continuous), in different shades of gray refer to a given number of registered approaches. As a matter of fact, the results confirm that on a relatively short time scale, a purely Keplerian reasoning could be sufficient. Having the semi-major axis a equal to the radius of the geostationary ring A, the inclined, eccentric orbits with the lines of nodes lying in the equatorial plane x ¼ 0 or x ¼ 180 will cross this plane at rmin ¼ að1 eÞ < A or rmax ¼ að1 þ eÞ > A. Let f0 ¼ arccosðeÞ defining the true anomaly of the point having r ¼ A. Then only the orbits with initial x ¼ f0 or x ¼ f0 þ 180 will exactly cross the geostationary
Fig. 2. The number of approaches closer than 250 km (top) and than 100 km (bottom) as a function of the eccentricity and the inclination of a geosynchronous orbit.
1196
I. Wytrzyszczak et al. / Advances in Space Research 34 (2004) 1193–1197
ring. For small eccentricities the values are close to x 90 and x 270.
4. Simulation 2: eccentricity–inclination plot The integration of the equations of motion for the second group (with common lines of apsides) was performed on the time span of 1 year, starting from T0 . In this case, the fictitious geosynchronous objects moved
on simple Keplerian orbits in order to simulate them as being active, maneuverable satellites. The real satellites, as previously, were subjected to all perturbing forces. Fig. 2 shows the number of approaches closer than 250 km (top) and 100 km (bottom) as a function of eccentricity and inclination of fictitious objects. For the reasons discussed in the previous section, only low eccentricities and, mostly, low inclinations can be responsible for a higher number of close approaches.
Fig. 3. The error in distance between the geosynchronous object and the 02029D geostationary satellite caused by an incorrect value of the geostationary satelliteÕs reflectivity coefficient.
Fig. 4. The error in distance between the same satellites as in Fig. 3, but resulting from an inaccurate value of the geostationary satelliteÕs area-tomass ratio.
I. Wytrzyszczak et al. / Advances in Space Research 34 (2004) 1193–1197
5. Error estimation Two possible sources of errors were estimated: incorrect values of reflectivity coefficients and incorrect values of S=m. Figs. 3 and 4 present the distance between the geostationary satellite 02029D and a geosynchronous object with e ¼ 0:007, i ¼ 28:7, x ¼ 0, X ¼ 45, after approximately one year of the simulated motion. In Fig. 3, the distance is plotted for three values of the geostationary satelliteÕs reflectivity coefficient: Cr ¼ 1, Cr ¼ 1:3 and Cr ¼ 1:6. The error in the distance of the two satellites after one year depends almost linearly on the adopted value of Cr . The difference DCr ¼ 0:1 generates in this case an error of about 3 km. Similar results, of the order of several km, were obtained for other satellites. Fig. 4 shows errors in satellitesÕ distance that result from neglecting the S=m variations caused by an aspherical shape of the object and its variable orientation with respect to the Sun. The same satellite 02029D was taken as an example. Depending on the orientation of the object, its S=m varies from 0.0044 to 0.0105 m2 /kg, with the mean value S=m ¼ 0:0089 m2 /kg. The error imposed by the variability of S=m amounts to about 20 km/yr.
6. Conclusions We were looking for optimal orbital elements set of a geosynchronous satellite in order to avoid close approaches to objects within the geostationary belt defined as a segment of a spherical shell A DR 6 r 6 A þ DR (DR ¼ 75 km) and delimited by 15 in latitude. Potentially most harmful objects move on orbits with perigees close to x 90 or 270. Every geosynchronous orbit of that type immediately crosses the GEO ring at the radial distance p ¼ Aðl e2 Þ if its eccentricity is less pffiffiffiffiffiffiffiffiffiffiffiffi than DR=A 0.042. The range of hazardous initial eccentricities shrinks as the direction of the line of apsides approaches that of the line of node. In the limit case, when x ¼ 0 or x ¼
1197
180, only orbits with e < DR=A 0.0018 can enter the GEO belt. The number of approaches varies a little in the range of considered GSO inclinations ð28 6 i 6 59Þ, slightly decreasing for larger inclinations. This conclusion apparently contradicts the previous results (Kessler, 1981; € Opik, 1951) that indicated the proportionality of the collision probability to sinði=2Þ= sin i. However, the classical collision rate estimates were developed under the assumption that the distribution of ‘‘target’’ objects is homogeneous in the volume of space at a given latitude. In the case of the GEO belt the number of collisions depends on individual contributions from all small layers between 15. Due to ‘‘librational’’ character of the relationship between orbital inclinations and longitudes of nodes (see for instance Wytrzyszczak, 2004) the density of nonactive geostationary objects falls down with latitude and is not homogeneous over geographical longitudes.
Acknowledgements The area-to-mass ratios data used in this paper were available thanks to the kind help of Prof. W. Flury and Dr. H. Klinkrad. The work presented in this paper was performed as a part of the KBN Grants number 8T12E 028 20 and 5T12D 026 23.
References Flury, W., Klinkrad, H., 2002 (private communication). Kessler, D.J. Derivation of the collision probability between orbiting objects; the lifetime of JupiterÕs outer moons. Icarus 48, 39–48, 1981. € Opik, E.J. Collision probability with the planets and distribution of planetary matter. Proc. Roy. Irish Acad. 54, 161–199, 1951. Wytrzyszczak, I. Testing the safety of decommissioned spacecraft above GEO. Adv. Space Res., this issue, 2004 (doi:10.1016/j.asr. 2003.02.036).