7.3 Solar pixel array model 105 a virtualxy-plane around thez-axis by azimuthφ. The resulting vector is then rotated around the virtual direction y’ by an angle ofθV1 =θ−π2. This can be expressed as:
V~1 =A′yAz
1 0 0
, (7.5)
whereA′y and Azare the quaternion rotation matrices for a rotation around y’ and z respectively.
Figure 7.1: Computation of vertex vectors for solar pixel array.
The second vertex vector V~2 can be computed from the vector product of ~rSUN and V~1. Now the vertexes are used for the generation of a solar plane perpendicular to the Sun-spacecraft vector where the plane centre is determined by the intersection of the plane and the vector from the satellite centre to the Sun~rSUN. For the resulting plane a total size has to be be specified. The projection of the plane onto the spacecraft has to fully engulf the surface model. For the preparation of the ray tracing (which is used for the detection of shadowed elements and the calculation of reflections) the solar pixel plane is partitioned into individual solar pixels. Here the pixel resolution determines the accuracy of absorption and reflection implementation.
The actual size of the individual solar pixelsxpixandypix is determined by the total number of lateral and longitudinal pixelsnxand ny and the specified total dimensions of the arrayxtot and ytot:
xpix= xtot
nx and ypix = ytot
ny . (7.6)
Now the centre position of each pixel within the array can be expressed as:
P~C(i, j) = (i−0.5)·xpix−xtot 2
·V~1+ (j−0.5)·ypix−ytot 2
·V~2−d·~xSUN, (7.7) where d is the modelling distance of the array surface to the spacecraft model and i andj denote the position of the pixel in the array as displayed in figure 7.2 left for the specific pixel (7,7).
Figure 7.2: Computation of individual pixel centre (left) and composition of complete solar pixel array (right) with~xSUN=−~rSUN.
Originating at each pixel’s centre, ray vectors pointing in the direction of −~rSUN are initialised as displayed in figure 7.2 right. Each ray is scanned for intersections with the model surface elements in order to detect illuminated and shadowed surfaces. For this the surface elements are sorted by distance from the current pixel centre and the hit evaluation (determined by a ray intersecting a surface element) is performed for the nearest elements first. If a hit is detected the respective ray is not traced further thus no other elements can be hit by the same ray. For each hit diffusely and specularly reflected rays are initialised as discussed in section. A received energy P(i) can be computed for each target surface intersected by a ray based on the mean solar flux P⊙(r) at the distance r, the surface of the target element A(i) and the surface normal direction~n(i):
Prec(i) =P⊙(r)A(i)~rSUN·~n(i). (7.8) Note that in this case the orientation vector is given byκabs = arccos(~rSUN·~n(i)). Fol-lowing the approach taken for the modelling of the TRP the SRP component resulting from absorption can be computed with:
P~SRP,abs(i) =−~rSUNα Prec(i)
m c . (7.9)
For specular reflection the resulting SRP component can be expressed as:
P~SRP,spe(i) =−~rSUNγsPrec(i)
m c +~xspe γsPrec(i)
m c , (7.10)
where ~xspe is the direction of specular reflection as modelled in section 3.3. Finally the SRP component resulting from diffuse reflection can be computed for each surface element by:
P~SRP,dif(i) =−~rSUNγdPrec(i) m c −2
3~n(i)γdPrec(i)
m c . (7.11)
7.3 Solar pixel array model 107 The resulting SRP can be calculated by the sum of all absorption and specular and diffuse reflection components of all individual surface elements:
P~SRP,res =
n
X
i
(P~SRP,abs(i)) +
n
X
i
(P~SRP,spe(i)) +
n
X
i
(P~SRP,dif(i)). (7.12) In this approach the resulting TRP vector and the individual surface TRP vectors are given in the spacecraft frame. The detection of radiation exchange between the different model surfaces is implemented with the ray tracing approach described in section 3.4.
Note that for multiple reflections the reflection contribution caused by a reflected ray which is absorbed again has to be subtracted fromP~SRP,spe(i). As for the TRP method, the SRP can be converted to a solar radiation force. If these forces are determined for each of the model surfaces and the centre of mass of the spacecraft is known a resulting solar radiation torque can be determined.
Chapter 8
The Rosetta Mission
This chapter gives an overview on the main characteristics of the Rosetta mission as well as the geometrical features of the spacecraft. The flyby anomaly is introduced and the motivation for SRP and TRP analysis for the Rosetta spacecraft is discussed.
8.1 Mission goals and trajectory
The international Rosetta mission is a cooperation between ESA, NASA and the na-tional European space agencies. The main scientific mission goal is the investigation of the solar system origin by examining the properties of cometary nuclei [78]. For this purpose Rosetta will rendezvous with comet 67P/Churyumov-Gerasimenko in 2014 where its lander payload Philae will land on the comet surface to study the physical and chemical structure of the body [79]. Other scientific goals are the examination of the evolution of the coma during the comets approach to the Sun, the interaction of solar winds with the satellite and a study of the two asteroids 2867 Steins and 21 Lute-tia which have already been passed on the way to the comet as displayed in figure 8.1.
In total Rosetta includes a set of 25 individual science experiments to be conducted in the various stages of the mission. In order to save fuel and to perform scientific experiments the satellite conducts a series of gravity assist manoeuvres, which partly have been tracked. Here the flybys of Earth are particularly interesting due to the observation of an anomalous increase of the spacecrafts velocity during the first flyby.
This so called flyby anomaly has been also been observed for other spacecraft and will be discussed in section 8.2 in more detail. In the times without manoeuvres the satel-lite performs so called cruise phases where attitude control is reduced to a minimum.
This creates ideal conditions for an observation of non-gravitational forces acting on the spacecraft.
Currently Rosetta is on its approach to Churyumov-Gerasimenko and has been put to deep space hibernation mode to ensure a minimum of energy consumption and hardware degradation. It will be put back to full operation in 2014 shortly before the final approach to the comet.
Figure 8.1: Rosetta Trajectory in heliocentric frame, 1: Start of Rosetta on 2004/03/02, 2: 1st Earth flyby on 2005/03/04, 3: Mars flyby on 2007/02/25, 4: 2nd Earth flyby on 2007/11/13, 5: Asteroid Steins flyby on 2008/09/05, 6: 3rd Earth flyby on 2009/11/13, 7: Lutetia flyby on 2010/07/10 8: Rendezvous with Churyumov-Gerasimenko on 2014/05/22, 9: Touchdown of Philae on 2014/11/10 [80].