Calculations are presented for the 1999 opposition class mission and a procedure for obtaining singlar occultation data for any other given Mars mission is given. Occultation data for a Mars orbiter in a 24.5 hour parking orbit and a Mars base were calculated for: sunlight occultation - the time in darkness; and radio communication occultation - the communication losses between the lander and the orbiter, the lander and Earth, and orbiter and Earth

Mulqueen, Jack

English

NASA Technical Reports Server

NANNEDNARS MISSION
SUNLIGBTAND CO_q_NICATION OCCULTATIONS
Jack Mulqueen
Marshal] Space Flight Center, AL
ABSTRACT
Calculations are presented for the 1999 opposition class mission and
a procedure for obtaining similar occultation data for any other given
Mars mission is given. Occultation data for a Mars orbiter in a 24.5
hour parking orbit and a Mars base have been calculated for: sunlight
occultation - the time in darkness; and radio communication occultation -
the communication losses between the lander and the orbiter, the lander
and Earth, and orbiter and Earth.
CALCULATIONS
Mars Orbiter Sunlthg_h__Occultation
To find the time in darkness for a Mars Orbiting Spacecraft it is
necessary to determine the orientation of the parking orbit with respect
to the Sun. This is done by finding the angle between the semi-major
axis of the orbit and the Mars to Sun llne. This angle, _ , (see
Figure 1), is found using the following equation:
x = RAP - VE - P - L
The values of RAP, VE, P and L are found using the trajectory data
(1) and the Planetary Handbook (2).
Once _ is known it is possible to find the points on the orbit
corresponding to the beginning and end of occultation, defined as_, and
This is done using the equation of the parking orbit and a
2
transformation between a reference frame centered on the orbit ( x, y)
and one that lies along the Mars-Sun llne (x" , y" ). These reference
frames are shown in Figure 2. The second frame, (x" , y ) is defined as
a rotation of the orbit centered frame (x, y) through an angle
followed by a translation d, where d is defined as:
d = ae sin
The quantity ae represents the distance from the center of the orbit
to the focus of the orbit which is the center of Mars.
The values of _ and _2 are found by finding the values of x"
that correspond to y" equal to _ the radius of Mars (RM), then con-
N87-1773 
142
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143
Z
0
t,,,,,
I-,-
-I
0
Of_
!
0
w_
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--Z
I.I=
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....I
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144
verttng to the x,y frame. [Once _1 and _ 2
equation then can be used to find the time from perigee to
The difference in the
occultation.
The transformation
the following manner:
First,
Xl 'Yl axes
are known,] Kepler's time
and1 2"
two times represents the duration of the
between the two coordinate frames are found in
a rotation of the x,y frame through an angle a to yield the
(see Figure 1).
x1 = x cos a+ y sin
Yl = -x sin _ + y cos_
Next, a translation along the y axis a distance d to yield the x'.
y axes.
x , y
x = x I = x cos _ + y sin
Y = Yl - d = -x sin a + y cos o._ d
or:
In matrix notation the transformation between the x,y frame and the
P
frame can be written as:
i: rco 1
To find the values of x"
the orbit must be found In terms of x
orbit in the x,y frame is:
x 2 + y2
Lsin C_cos _] Ly + d
corresponding to y = +_ RM the equation of
and y The equation of the
= 1
or:
x2b 2 + y2a2 - a2b 2 = 0
using the transformation:
x = x cos a - (y + d) sin
y = x" sin _ + (y ÷ d) cos
145
The equat ion of tile orbit in terms of x " and y is:
., [ b 2 ? . . a 2 b 2)x • cos 2 (x + a2sin E_, ] , x [2(Y -_ d)( - sin _Zcos _z ]
+ [(y" d) 2 (b2sin 2 (_ + a2cos 2 (x) - a2b 2] = 0
Given y" = _+RM the corresponding values of x" can be found using
the quadratic formula, then these values of x" and y" can be converted
to the x,y frame using the transformation matrices. The values of _b and
1
are found using the x _md .V coordinates of the orbit at the beginning&
2
end of occultation.
--1
= tan ' y
1_ I I -x- beginning
2
The duration of the occultation is found by
equation to calculate the lime from perigee at _b
1
t ( _-e sin _)
P ?-
The duration of the occultation Is:
end
using Kepler's time
and
2
A t t - t
P P
2 1
The orbiting spacecraFL is occulted for a period of A t once during
orbit. The value of A t changes during the staytime as the longi-each
tude of Mars changes and the orbit precesses. To obtain the minimum and
maximum occultations, 5 t should be calculated on Mars arriva] and depar-
ture.
Mars Lander Sunlight Occultation
The time in darknes:_ that a Mars ]ander would be subjected to is
highly dependant on the latitude of the landing site and the heliocen-
tric longitude of Mars. The amount of daylight varies on Mars just as it
does on Earth, since its equator Is Inclined to Its orbit by 23.984
degrees. To calculate the time in darkness, the following calculations
are required.
The geometry shown in Figures 3a and 3b represents the orientation
of the axis of rotation of Mars with respect to the Sun. n represents
the angle between the polar axis and the vertical as viewed perpendicular
to the Mars-Sun line. Using Figure 3a, n can be found:
146
Z
m
r_
Z
_L
W
_rn"
_0
i
u.>-
n.,
F-
W
0
W
A
-r
F,-
<
f,_ w
,,- zo _o
v
Z W
0 .I-"
I*- F"--_
n- <x
< a.<
IL N-
m
\
ul
.J
CJ
0 <
Z
_o
CJ
f-
Z<
_w
c_n,.
k_
147
J = 23.984 DEG
r = RN sln I
y = N cos L = RN sin I cos L
z = RN cos I
Figure 3b shows the orientation of Mars wlth respect to the Sun and
a cross section of a particular latitude shows how much of that latitude
Is in the sunlight. The tlme In darkness Is found using the following
calculations and Figure 3b.
q =
z
tan n =
I RN sin _ IX
tan n
c°s-I [L_-----'-I= c°s-I
2q = 2 cos -1
I sin _ 1cos n
sln _ tan
cos L
2
24.5 hrs
Communication Occultatlon Between The Mars Orbiter And Earth
Thls calculation Is performed using the same procedure as the Mars
orbiter sunlight occultation except instead of a being used In the
calculations, a different angle, _ Is used. B Is defined as the angle
between the seml-major axls of the parking orbit and the Mars to Earth
line. The following calculations are required (see Fig, 1).
r I_ (1-e2)e + e_osel Earth
148
r
m
J = tan
÷ e
X = r -
m r e cos Q
Y = r e sin Q
--ff-
_=u-J
Mars
ter
the
degrees.
assumed
perigee.
orbiter.
degrees.
Given a, e,p and _.
Once B is found the procedures for finding the communication occultatlons
are identical to the sunlight occulatlons. Starting with equation 1, Bls
substituted for _ .
Communication Occultation Between The Mars Lander and Earth
The communications occultation between the Mars lander and the Earth
is dependent on the same parameters that influence the sunlight occulta-
tion. Since the plane of the Earth's orbit Is inclined only 1.849
degrees to the Mars orbit plane, the value of the duration of the
communications occultation would be essentially equal to the duration of
the sunlight occultation for a given latitude.
Communication Occultation Between The Mars Lander and The Mars
Orbiter
The communication occultation of the Mars lander and the Mars orbl-
Is obtained by finding when the angle between the local vertical at
landing slte and the position of the orbiter Is greater than 90
The geometry for thls calculation Is shown in Figure 4. It was
that the orbiter Is directly above the lander when It Is at
The angle c represents the angle between the vertical and the
Communication Is occulted as long as c Is greater than 90
Using Figure 4, c was obtained wlth the following calculations.
t = tp = _-_ ( $ - e sin $ )
= 2_ PAD
24.5 HRS
149
>.
W
0
Z
0
I
Z
=8
a:
0
I
Z
.J
Q:
I!
n"
n-
O
150
_=2_-_t NOTE : Is equal to thls only when the parking
orbit is retrograde, i.e., direction Is opposite
Mars's rotation.
h= + r -2RNr
-1
k = sln
c=w-K; h>r
cos J
c = K ; h<r
DATA FOR BASELINE, 1999 NISSION
24.5 hour parking orblt
r = 3900 km
P
a = 20364.63 km
e = .8084915
MARS ORBITER SUNLIGHT OCCULTATION
MARS ARRIVAL MARS DEPARTURE
RAP = 273.816 deg 275.242 deg
VE = -67.01 deg -67.01 deg
P = 335.323 deg 335.323 deg
L = 168.915 deg 195.73 deg
= 16.588 deg 8.801 deg
AT = .267 hr (16.06 mln.) .264 hr (15.85 mln.)
NARS LANDER SUNLIGHT OCCULTATION
No landlng
Nars day,
r a = 36829.2 km
slte has been baselined s_ assume occultatlon of 1/2
CONNUNICATION OCCULTATIONS
ORBITER TO EARTH
MARS ARRIVAL
L = 168.91
LE = 115.
Q = 53.91
@ = 12.747E
At = 12.25 hrs
151
NARS DEPARTURE
195.73
176.
19.73
73.477
@N = 193.588
r = .98367 AU
e
r = 1. 6604 AU
m
X = 1.0809 AU
Y = 1.315 AU
J = 50.59 deg
= 34. 002 deg
AT = .2823 hr (16.93 rain.)
LANDER TO EARTH
220.41
.99498 AU
1.62526 AU
.68449 AU
.33589 AU
26.138 deg
17.337 dee
.2677 hr (16.06 mln.)
Since no landing site has been chosen,
tatlon occurs for half of the Mars day.
AT = 12.25 hr
LANDER TO ORBITER
These calculations were made by calculating c for _ =
The results are shown below:
assume communication occul-
tp (hr) _(deg) c (deg)
0 0 0
.3 18 79.2
.4 24 108.7 occultation begins
7.5 140 89.2 occultation ends
12.25 180 0
17.0 220 89.2
17.2 222 93.2 occultation begins
24.1 336 108.7 occultation ends
degrees,
AT = 7.1 hrs occurlng twice every orbit.
0 to 360
REFERENCES
1. Young,
,
A., "Mars Mission Concepts and Opportunities";
MSFC paper in Section II.
Planetary Fllght Handbook, Volume 3, NASA SP 35 Part 1,
1963.
152
LIST OF SYMBOLS
a = semi-major axis of an orbit
b = seml-mlnor axls or an orbit
c = angle between local vertical at landing sight and the Mars orbiter
d = distance from origin of x,y reference frame to the origin of the
Xl 'Yl frame
e = orbit eccentricity
i = inclination of Mars' equator to the ecliptic
I = latitude of landing slte
L = heliocentric longitude of Mars
LE = heliocentric longitude of,Earth
P = heliocentric longitude of Mars perihelion
RM = radius of Mars
RAP = right ascension of perigee of Mars parking orbit
r = distance from center of Mars to orbiter
r = distance from center of sun to center of Earth
e
•r = distance from center of sun to center of Mars
m
tp I = time from perigee that occultation begins
tp2 = time from perigee that occultation ends
At = duration of occultation
VE = heliocentric longitude of Mars vernal equinox
x,y = coordinates in the x,y reference frame (Mars parking orbit)
x_ , y" = coordinates in the x _ , y " reference frame
= angle between seml-major axis of parking orbit and Mars-Sun llne
= angle between semi-major axis of parking orbit and Mars-Earth line
= eccentric anomaly
n = angle between vertical and the Mars polar axis as seen perpendicular
to the Mars-Sun llne
= gravitational constant km3/sec 2
0 = true anomaly
153


Manned Mars mission sunlight and communication occultations

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