MAE 5410 – Astrodynamics Lecture 5 Orbit in Space Coordinate Frames and Time.
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Transcript of MAE 5410 – Astrodynamics Lecture 5 Orbit in Space Coordinate Frames and Time.
Orienting the orbit planeSo far, we’ve solved for the orbital motion in the orbital plane (PQW) which is given by the following parameters that can be calculated from a position and velocity at any epoch time
Now we’ll orient the orbit plane (i.e. PQW) in space using three angles. Since the orbit is inertially fixed, we use the Earth Centered Inertial frame as a reference.
location
shape
size
),(),(),(
,,
tMtEtf
e
a
tvr ooo
ECI: The X-Y axes are the the Earth’s equatorial plane, with X pointing along the intersection of the equator and the ecliptic (vernal equinox or line of Aries) direction. Z is along the Earth spin axis.
These directions change ever so slightly (Earth precession has 26,000 year period with a 18.6 year 9 arcmin nodding) so the vernal equinox direction at a particular time is used as a standard. Right now, J2000 is the standard reference. In 2025, we’ll switch to J2050.
Inclination, iAngle between the orbit plane and the equatorial plane
Z
YX
Increasing the orbital inclination increases the maximum latitude of the groundtrack (in fact, the maximum latitude equals the orbit inclination)
Longitude of the Ascending Node, Angle between the X-axis and the intersection of the orbit plane and
equatorial plane (the nodal vector)
Z
Y
X
Argument of Perigee, Angle from the nodal vector to the periapsis point (eccentricity vector, or )
#1 satellitefor P
#2 satellitefor P
P
r(t) and v(t) in ECIIn Lecture 3 we found the position and velocity in the PQW frame:
qEepEEe
aμv
qEeapeEar
ˆ cos1ˆ sin)cos1(
/
ˆsin1ˆ)(cos
2
2
In this lecture we defined orbital elements that locate the PQW frame wrt the ECI frame.q
w
p
w
q
p
k
j
i
w
q
p
k
j
i
v
v
v
T
v
v
v
r
r
r
T
r
r
rTo get from PQW to ECI, we perform a coordinate transformation:
f
Single Axis Rotations
R
R
R
Rot
B
B
B
Z
Y
X
θθ
θθ
Z
Y
X
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
1
11
11
cos sin- 0
sin cos 0
0 0 1RZ
RX
RY1
1
BX
BYBZ
R
R
R
Rot
B
B
B
Z
Y
X
θθ
θ
Z
Y
X
ˆ
ˆ
ˆ
10
0
ˆ
ˆ
ˆ
2
22
2
cos0sin
0
sin-cos 2
R
R
R
Rot
B
B
B
Z
Y
X
θθ
θ
Z
Y
X
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
3
33
3
100
0cossin-
0sincos 3
RX
RY
RZ2
2
BY
BZBX
RY
RZ
RX3
3
BZ
BXBY
Transformation from ECI to PQWFirst do a three axis rotation of , then a one axis rotation of I, then a three axis rotation of :
qw
p
k
j
i
k
j
i
w
q
p
r
r
r
iii
iii
iii
r
r
r
ii
ii
r
r
r
cossincossinsin
cossincoscoscossinsincoscossinsincos
sinsinsincoscoscossinsincossincoscos
1
100
0cossin-
0sincos
cos sin- 0
sin cos 0
0 0 1
100
0cossin-
0sincos
r(t) and v(t) in ECFTo get from PQW to ECI we invert the previous transformation, which turns out to just be the transpose:
To get from ECI to ECF we rotate through the Greenwich mean sidereal time:
w
q
p
k
j
i
r
r
r
iii
iii
iii
r
r
r
coscossinsinsin
sincoscoscoscossinsinsincoscoscossin
sinsincoscossinsincossincossincoscos
ECIECFECIECF vTvrTr ,
100
0)cos()sin(
0)sin()cos(
GSTGST
GSTGST
TECF
Greenwich meridian
GST
ECI
r(t) in SEZTo get from ECF to the topocentric-horizon frame, SEZ, we rotate through latitude, , and longitude, and subtract off the position vector to the site on the Earth:
sinsincoscoscos
0cossin
cossinsincossin
T
earth
ECFSEZ
R
rTr 0
0
ECF
SEZ
This vector can then be used to find the azimuth and elevation of the satellite with respect to the observer on the ground