The ballistic support of the

1
The ballistic support of the SPECTR-RG
spacecraft flight to the L2 point of the
Sun-Earth system

• I.S. Ilin, G.S. Zaslavskiy, S.M. Lavrenov, V.V.
  Sazonov, V.A. Stepaniants, A.G. Tuchin, D.A.
  Tuchin,
• V.S. Yaroshevskiy
• Keldysh Institute if Applied Mathematics RAS
• 2012

2
The quasi-periodic orbits in the vicinity of the
L2 point of the Sun-Earth system
3
Missions to the L2 point of the Sun-Earth system

• Two Russian missions are to be sent to the
  vicinity of the L2 point during the next few
  years
• The Spectr-RG spacecraft, flying to the L2
  point of the Sun-Earth system and staying at the
  halo orbit in its vicinity. NPO S.A. Lavochkina,
  2015.
• The Millimetron spacecraft, flying to the L2
  point of the Sun-Earth system and staying at the
  halo orbit in its vicinity. The spacecraft has
  to go out far from the ecliptics plane. NPO S.A.
  Lavochkina, 2018.
• The examples of the L2 point missions that have
  already been implemented
• NASA spacecraft WMAP, (2001 2009)
• ESA spacecraft Planck space observatory
  Hershel (2009)
• ESA space observatory spacecraft Gaia should
  go to the vicinity of the L2 point of the
  Sun-Earth system in 2013

4
The Spectr-RG mission

• The Spectr-RG mission presupposes the flight to
  the vicinity of the Sun-Earth system L2 point and
  the halo orbit motion in the L2 point vicinity
  during the 7 years period.
• The halo orbit in the vicinity of the Sun-Earth
  system L2 point is opportune because of the
  possibility of reaching it with a single-impulse
  flight with no correction at its end.
• To keep the spacecraft in the halo orbit the
  stationkeeping is needed. Total stationkeeping
  costs for the 7 years period must not overcome
  200 m/sec.

5
The isoline of the pericentre height function
building method.

• The isoline method for the approximate
  description of the Earth L2 trajectories was
  suggested in M.L. Lidovs papers. It was applied
  for the direct single-impulse flights without any
  Lunar swing by maneuver.
• The spacecraft motion is described in the
  rotating reference frames in the geocentric
  reference frame and in the reference
  frame with the beginning in the L2 libration point

The average values of A(t) ? B(t) are chosen at
the halo orbit designing stage. The average value
of C(t) must be close to 0.
6
The linearized equations of the spacecraft motion
in the quasi-periodic orbit in the rotating
reference frame
7
The integration constants


µ1, µ the Sun and the Earth gravitational
constants
rL1, rL the distances from the L2 point to the
Sun and the Earth
a1 the astronomical unit
n1 the average angle speed of the Earth orbital
motion.
8
The isoline building algorithm

• The search of the pericentre height function
• according to the
  following algorithm
• The spacecraft state vector is calculated in the
  inertial reference frame, obtained by the
  fixation of the rotating reference frame axes at
  a fixed moment of time according to the
  parameters ?, B, and .
• The obtained state vector is converted into the
  non-rotating geocentric ecliptic reference frame.
  
• The geocentric orbit elements are counted with
  the help of the obtained state vector with the
  pericentre height among them.
• The first isoline dot search
• The extension of the isoline to the next dot

9
The first isoline dot search
The scanning is performed within the interval
from 0 to 360 for f1 and within the interval
from 180 to 180 for f2 with the step of 45º
for f2 and 1º for f1 .
The f1 value satisfying the following condition
is looked for
With the help of the bisection method the f1m
value satisfying the following condition is
searched
The pair of f1m, f2 values found is the isoline
beginning point.
10
The extension of the isoline from the current
point
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The examples of the obtained isolines
The isolines within the 27.01.14 launch window
with the Moon swing by maneuver
The isolines within the 18.12.14 launch window
with the Moon swing by maneuver and 1 lap at the
LEO
The isolines without the Moon swing by maneuver
f2
f2
f2
f1
f1
f1
from 0.18 to 0.2. 0.1
12
The structure of the nominal transfer trajectory
calculation algorithm

• The isolines built are the income data for the
  flight trajectory initial kinematics' parameters
  calculation algorithm the initial approximation
  of the transfer to the halo orbit.
• The initial approximation built is used for the
  exact calculation of the flight from the Earth
  orbit with the fixed height to the given halo
  orbit. The kinematics' parameters vector is
  counted more precisely according to the edge
  conditions.
• The velocity impulses, needed for the
  stationkeeping of the spacecraft in the given
  area around L2 point are counted.
• The shadow zones and radiovisibility zones for
  the locating stations, situated on Russian
  territory are counted for the whole spacecraft
  lifetime.

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The initial approximation calculation. The
transition from the transfer trajectory to the
halo-orbit
The condition to select the one impulse
transfer trajectories
With the fixed A, B ? C 0 an isoline is build
in the f1, f2 plane
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The stages of the nominal trajectory calculation

1. The velocity vector of the hyperbolic transfer
   trajectory, obtained from the initial
   approximation is counted more precisely according
   to the edge conditions which are the given values
   of the parameters B and C 0.
2. The velocity vector, obtained at the stage 1 is
   counted more precisely according to the condition
   of the maximum time of the halo-orbit staying in
   the L2 area of the following radius

15
The calculation of the stationkeeping impulses,
keeping the spacecraft in the halo orbit in the
L2 area
,
,

- the partial derivatives of the FC function with
respect to the components of the velocity vector
- the biggest possible value of the impulse
- the coefficient, controlling the step decrease.
16
The isoline method for the Moon swing by
transfers
It is opportune to use a Moon swing by maneuver
for the halo orbit transfer trajectories, as it
allows to find the orbits coming closer to the L2
point.
For calculations of the pericentre height
corresponding to the given halo orbit the
trajectory is divided into 3 parts

• the flight from Earth to the entrance into the
  Moon incidence sphere,
• the flight inside the Moons incidence sphere,
• the flight after leaving the Moons incidence
  sphere till the entrance of the L2 point vicinity.

For searching the pericentre height these parts
of trajectory are passed backwards. The function
of the pericentre height also depends on time in
case of the Moon swing by maneuver being applied.
17
The transfer trajectory without the Moon swing by
maneuver
The XY plane view, the rotating reference frame,
mln. km.
18
The transfer trajectory with the Moon swing by
maneuver
The XY plane view, the rotating reference frame,
mln. km.
19
The transfer trajectory with the Moon swing by
maneuver and the preliminary lap at the LEO
The XY plane view, the rotating reference frame,
mln. km.
20
The XY, XZ, YZ plane views of the halo-orbit in
the rotating reference frame. The transfer to the
halo-orbit is performed with the help of the Moon
swing by maneuver
Dimension thousands of km
200
200
500
-200
1500
-200
1500
500
-500
The total characteristic velocity costs for the
stationkeeping are about 30 m/sec for the 7
years period.
21
The XY, XZ, YZ plane views of the halo-orbit in
the rotating reference frame. The transfer to the
halo-orbit is performed with the help of the Moon
swing by maneuver. There was 1 preliminary lap at
the LEO.
Dimension thousands of km
200
200
500
-200
1500
-200
1500
500
-500
The total characteristic velocity costs for the
stationkeeping are about 30 m/sec for the 7
years period.
22
The halo-orbit, calculated for the Millimetron
project. The XY, XZ, YZ plane views in the
rotating reference frame
Dimension thousands of km
900
1100
900
-1100
1500
-1100
1500
-700
1500
The total characteristic velocity costs for the
stationkeeping are about 14 m/sec for the 7
years period.
23
The evolution of the orbit parameters ,
and
t , days
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The transfer to the L2 vicinity with the help of
the Moon swing by maneuverThe dates of the
transition to the L2 vicinity for 2014 year
Month ?A Launch date The duration of the launch window, hours
January 0.14 20140128 36
  0.15 20140128 72
February 0.14 20140227 40
  0.15 20140226 48
March 0.12 20140329 46
April 0.12 20140427 24
May 0.12 20140529 20
  0.13 20140529 28
  0.14 20140529 36
  0.15 20140529 52
June 0.12 20140625 22
  0.13 20140625 33.5
  0.14 20140625 40.5
  0.15 20140625 60
July 0.15 20140725 41
August 0.14 20140824 14.5
  0.15 20140823 57.5
September 0.12 20140922 12.5
October 0.12 20141023 6
November 0.12 20141121 23.5
December 0.12 20141218 22
25
The contingencies for the Specter-RG spacecraft
orbit

• To provide the needed level of solar cell panels
  luminance and radiovisibilty conditions for the
  Russian tracking stations, the following
  circumstances were taken into account at the
  orbit design stage
• If the spacecraft comes too close to the ecliptic
  plane, the penumbra area entrance is possible
• If the spacecraft goes too far from the ecliptic
  plane, long periods of no radiovisibility are
  highly probable.

26
The results of the research

• The ballistic problem of obtaining halo orbits
  with the given geometric dimensions in the
  ecliptic plane and in the plane orthogonal to it
  has been solved.
• A new method of transfer trajectories building
  for the flight from LEO to the family of halo
  orbits in the vicinity of the Sun-Earth system L2
  point is developed. These trajectories need no
  impulse for the transfer from the flight
  trajectory to the halo-orbit.
• The stationkeeping velocity costs are evaluated.
• The primary evaluations of the orbit parameters
  determination and the forecast accuracy have been
  obtained.