ON THE THEORY AND SOME SPECIFIC METHODS FOR SEARCHING FOR SPACE OBJECTS BY USE OF ROUGH ORBITAL INFORMATIONS. Veniaminov, V. Zavaliy, Yu. Tretyakov, V. Pyrin, V. Lapukhin
Scientific Research Center "Kosmos", Moscow
At different conferences, symposia, and workshops the topics on positional observations of space objects (SO) are usually concerned with two marginal categories only – the survey and the tasked observations. The same situation took place at the 4th European Conference on Space Debris held in Darmstadt, April this year.
The traditional approach in the first case means the consecutive scanning of some fixed space area and detecting everything one can find there, the signal energy of a moving SO being smeared on the receiver which does not help the acquisition of a weak signal, if so.
Besides, this strategy has two more disadvantages. These are mistakes of the 1st and 2nd kind – that is the availability of "slits" in the search plan through which the search for SO can “come down”, and the significant redundancy of the search plan elements overlapping in terms of what is necessary and sufficient.
All this implies unreliability of detecting SO and non-economical expense of the sensor resources, especially when there is the task of detecting given specific SO.
The second case is the second extreme. It needs very accurate ephemerides. But what if there are only rough ephemerides?
At the same time, between these marginal cases there exists one practically very important type of the search situation, namely, the case when one needs to detect the specific SO by use of a somewhat rough a priori orbital information. Such a search situation arises in space surveillance practice, for instance, when the initial information on the search for SO is available: the data on the time and place of the launch; the designed orbital data; the statistical orbital data on a specific class of SOs; some estimates of the orbital parameters by rough radar or optical measurements; some coverage information on the SO orbit; some information on the maneuver or the orbital correction; the information on the orbital structure of a constellation; and so on, and so on.
When we face the search situation of this type it is inexpedient to address the traditional scanning strategy because of its unreliability and inefficiency for this kind of tasks. The second approach mentioned fails because of lack of accurate orbital data.
By the way, mathematically, neither the first, nor the second category of the task is interesting. Every task in its general setting has two trivial extremes in which it degenerates mathematically: 1) when all the parameters are absolutely unknown, and 2) when the values of all parameters are given. The task is mathematically interesting if only some parameters of the full set are known or their values are roughly given.
Given the rough a priori orbital information, the search problem has some mathematical intrigue. It is clear that if one disposes only of the rough orbital information, one should pay some additional expense of the sensor resource for detecting the SO. But one should find the solution to make this additional expenditure minimum.
The difficulties in getting the optimum solution arise from the necessary condition of decomposing the continuously drifting and topologically deforming real SO current position uncertainty domain into the search plan elements rather than into some fixed space area (as in the simple scanning case).
Still more complication of the task originates from the fact that this 3-dimensional domain (for instance, the uncertainty ellipsoid) is projected onto the picture plane of the sensor used (also, by the by, drifting and deforming). As a result the structure and the mutual position of the already inspected elements and the ones still to be inspected becomes topologically very complex (see Fig. 1).
If at time-moment t0 one “cuts” the uncertainty ellipsoid projection into the search plan elements in conformity with the size and form of the sensor field of view, then in some time at time-moment t1 the full picture will nonlinearly change. The element A inspected at time- moment t0 will partially “come crawling” across the neighboring element B still to be inspected at time-moment t1, and vice versa.
This occurs because a point of the uncertainty domain projection at time-moment t0 transforms to a set (rather than a point) of the uncertainty domain projection at time-moment t1.
For the constructive analysis and creating the real search plans with due regard for all considered processes, a special search theory was developed [1, 2, 3] based on the so called search plan equivalence principle [4, 5].
Fig. 1. The peculiarity of the temporal error transformation when projecting from the real uncertainty domain onto the sensor picture plane.
In terms of this theory the search problem constructively comes to choosing a sequence of conditional ephemerides and related conditional velocities of the SO's supposed motion. This sequence is to be optimum with respect to the adopted efficiency criterion under the restrictions laid by technical capabilities of the sensor used.
This set of conditional ephemerides and velocities (so called the generalized ephemerides ) should completely cover not only the area of space, but also the continuously drifting and topologically deforming SO's current position uncertainty domain. Besides, this sequence (the search plan) should be the most economical one.
Eventually, after realization of this plan the entire complex of necessary favorable conditions for correctly targeting the sensor, and keeping this direction for concentrating and accumulating the wanted intelligence signal energy in one point of the receiver, must be provided in one and only one case – that is for only one generalized (conditional) ephemeris of the set. And that will do for solving our problem. All other conditional ephemerides area waste, but the necessary and minimum waste, the necessary and minimum payment for the roughness of the a priori information about the search for SO.
Significant simplification of the task and related transformations, and reduction of degrees of freedom can be achieved in the following, practically important cases: the search for a lost SO; the search for a SO for which there are no measurements for a long time; the search for a SO moving in a highly elliptical orbit after its detection by a radar in its perigee zone; the search for a SO by a very narrow-angled facility. In these and many other cases one may neglect all the errors but those along the track. This circumstance crucially reduces the problem dimension in terms of the equivalence principle.
Such cases form an important niche in the Space Surveillance practice. For such cases some search programs were developed in terms of the theory above, which were implemented at some sites, for example, at the “Okno” site in Tajikistan. They realize the discrete search optimum plans by the argument of latitude u. Two of them are the programs beginning from the edge of the search area in different directions. The third one begins the inspection of the search area from its calculated center. One can find more details in .
All the programs were tested together with the traditional approach (the consecutive scanning of the fixed space area). The comparative results are given in the Table 1 where 1 is the search program beginning from the edge of the search area in direction of the SO motion; 2, the search program beginning from the edge of the search area counter the direction of the SO motion; 3, the search program beginning from the calculated center of the search area; 4, the traditional method; Δts, the real duration of the search; CO, the 12-hour circular orbit; HEO, a highly elliptical orbit.
Due to results of the work of programs the average economy of the facility resource is about
where Ttr is a standard resource spent by the traditional method for detecting one SO; N, full number of the searches for SOs in the group. One can see from the Table that in 4 cases out of 10 the SO could not detected by use of the traditional method.
One more important effect of using the above search technology is the possibility of detecting small and weakly-contrasting space objects.
Table 1. Results of testing the traditional and proposed search methods
Размещен 29 ноября 2006.