Fragmentation profile assessment

This section introduces an algorithm to estimate the∆Vgiven to each fragment at the breakup epoch and a coordinate system to be applied to the estimation. The fragmentation profile of the 1968-081E breakup will be assessed based on the∆Vestimation result.

Fragments are scattered from the parent object at the time of the breakup, thus this situa-tion can be translated as follows; the orbits of the parent object and its fragment shares a same inertial point at the breakup epoch, and the∆Vis given to the fragment at the inertial point. In actual situation, however, one cannot specify the inertial point shared by the parent object and its fragment even at the breakup epoch due to uncertainties caused by orbit determination errors and orbit propagation errors. Thus,∆V of each fragment should be estimated at each proba-ble breakup point, where the distance between the parent object and the fragment becomes minimum. This study applies a close approach analysis to find the closest points between the parent object and each fragment at the breakup epoch. The Nelder-Mead method (Nelder &

Mead, 1965), which is a well-known extremum-seeking algorithm, is applied to find closest points between each fragment and the parent object at the breakup, i.e., to derive a proper∆V for each fragment. Thus, the orbital states of each fragment and the parent object are required for the analyses so that this study uses the latest TLEs of the 22 fragments catalogued in SSN as of January 2013, including 1968-081J, K, L, M, N, P, Q, R, S, T, U, V, W, X, Y, Z, AA, AB, AC, AD, AE, and AF. Orbits of the parent object and its fragments are back propagated to the breakup epoch of 1968-081E, and then a probable breakup point for each fragment is estimated by conducting a close approach analysis between the parent object and each frag-ment. The orbit propagation adopts special perturbation theories including perturbation forces by non-spherical part of the Earth gravity (by applying EGM96 to the 8th order), third bod-ies (the Sun and the Moon), and SRP. A/m of 0.01 m²/kg is assumed for each object. Figure 5.1 shows an example of the close approach between the parent object and a fragment. It is demonstrated that the extremum-seeking algorithm is capable of finding the global minimum of the distance between the parent object and the fragment.

In this section, ∆V vector components of each breakup fragment are discussed in terms of the satellite coordinate system NT W (Vallado, 2001). Figure 5.2 schematically illustrates the relation between a parent object and its breakup fragment in theNT Wcoordinate system.

The NT Wcoordinate system is centered at a spacecraft, whereN axis is on the orbital plane

5.1 Fragmentation profile assessment

Figure 5.1: A demonstration of the close approach between the parent object and a fragment.

- The minimum distance detected by the close approach algorithm is represented as the red point.

The initial point where the close approach starts is centered in the figure. The histories of the minimum seeking is depicted as the red line.

and directs outward from the Earth (in-plane),T axis points to a velocity vector direction of the spacecraft in the geocentric inertial frame (in-track), andWaxis is orthogonal toNandT axes with the right-hand rule (cross-track). TheNT Wframe can describe the relative position and relative motion of an object with respect to the object centered at the frame. This study defines the velocity vector components of a fragment in aNT Wframe as the∆V vector of the fragment.

T (in-track)

W (cross-track) N (in-plane)

Azimuth

Elevation

Breakup fragment Parent object ΔV vector

Figure 5.2: Relation between a parent object and its breakup fragment in theNT W frame.

To begin with discussions of the estimated ∆V distribution, this study confirms that the 22 catalogued fragments are good for representing the fragmentation profile of 1968-081E.

For this study, it is verified whether or not observation point selection in survey observations causes biased characteristics in the∆V distribution of breakup fragments acquired by the ob-servations. To evaluate this hypothesis, four quadrants are defined in the ∆V direction, i.e., azimuth-elevation, as specified in Figure 5.3. This study generates 1968-081E fragments by using the NASA standard breakup model 2001 revision with the scaling factor of 1.0, and divides them into 4 groups with respect to the quadrants. The NASA breakup model is imple-mented to assume that the breakup fragments are released into every direction with a uniform probability at the breakup, so that each group of the fragments represents characteristics of the quadrant. Figure 5.4 compares the predicted populations of 1968-081E fragments generated by the NASA breakup model in each∆Vquadrant at the breakup. The time-averaged distribution evaluated by 100 MC runs is applied to the OD generation and orbit propagation processes to represent each population. Fragments detection can be expected at the regions emphasized by the orange gradient, though there is no significant difference between each quadrant population other than the foot regions colored in gray. As an approximation, the actual population can

5.1 Fragmentation profile assessment

be regarded as a superposition of each quadrant population. Thus, it can be concluded that no biased characteristics in the∆Vdistribution are not induced by the observation points selection in survey observations.

The evaluated ∆V distribution of the 22 fragments, which represents the fragmentation profile of the 1968-081E breakup, is shown in Figure 5.5 as an orthographic projection of the NT W frame. The subframe illustrated in each plane represents the ∆V caused to the after-breakup parent object by the before-after-breakup parent object at the after-breakup epoch. Most of the fragments have∆V smaller than 100 m/s, whereas the maximum∆Vin the data set is 195.106 m/s. It is obvious that the fragments∆V distribution is represented by one direction and its reverse direction. What is more,∆V vector of the parent object points to one of the directions.

From these features, it can be hypothesized that the fragments were scattered from one side and its opposite side of the Transtage 1968-081E at the breakup. This hypothesis is consistent with the shape of the Transtage 1968-081E estimated by inverse photometric analysis, as reported in (Rykhlovaet al., 1997). Usually, isotropy is assumed for∆V direction modeling of a breakup event, but anisotropy can be confirmed in the breakup event of 1968-081E.

2nd Quadrant 1st Quadrant

3rd Quadrant 4th Quadrant

Figure 5.3: The definition of four quadrants in the azimuth-elevation plane.

Figure 5.4: Comparison of 1968-081E populations generated by the NASA breakup model in each∆V quadrant. - The epoch of each population is set to be 21 October 2011. The scaling factor of 1.0 is assumed in the OD generation. 100 MC runs are applied to represent the resulting populations.

5.1 Fragmentation profile assessment

T W N

T N W

N W

T

Figure 5.5: The∆V distribution of 1968-081E fragments in theNT W frame of the parent object at the breakup epoch.- A vector plot in the sub-frame indicates the∆Vchange caused to the parent object.