Transcript MAE 5410 – Astrodynamics Lecture 1A - AGI

MAE 5410 – Astrodynamics
Lecture 5
Orbit in Space
Coordinate Frames and Time
Orienting the orbit plane
So 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
a


ro , vo , to  
e
 f (t ), E (t ), M (t ),

size
shape
location
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.
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, i
Angle between the orbit plane and the equatorial plane
Zˆ
Increasing the orbital inclination
increases the maximum latitude of
the groundtrack (in fact, the
maximum latitude equals the orbit
inclination)
Xˆ
Yˆ
Longitude of the Ascending Node, W
Angle between the X-axis and the intersection of the orbit plane and
equatorial plane (the nodal vector)
Zˆ
Xˆ
Yˆ
Argument of Perigee, w
Angle from the nodal vector to the periapsis point (eccentricity vector, or Pˆ )
Pˆ for satellite#2
Pˆ for satellite#1
Putting it all together
Some special cases
f
l
f

u
r(t) and v(t) in ECI
In Lecture 3 we found the position and velocity in the PQW frame:
r  a(cosE  e) pˆ  a 1  e 2 sin Eqˆ
v
wˆ

μ/a
 sin E pˆ  1  e 2 cos E qˆ
(1  e cos E )

In this lecture we defined orbital
elements that locate the PQW frame
wrt the ECI frame.
qˆ
f
pˆ
To get from
PQW to ECI,
we perform a
coordinate
transformation:
 ri 
rp 
r   T  r 
 j
 q
rk 
rw 
 vi 
v p 
v   T  v 
 j
 q
vk 
vw 
Single Axis Rotations
Zˆ B
Zˆ R
 Xˆ B  1
0
0   Xˆ R 
 ˆ  
 ˆ 

 YB   0 cosθ1 sinθ1   YR 
 Zˆ  0 - sinθ cosθ   Zˆ 
1  R 
 B  1
YˆB
1
1
Xˆ B
Xˆ R
Xˆ B
YˆR
Rot1
Xˆ R
Zˆ B
2
2
YˆB
YˆB
YˆR
YˆR
3
Zˆ B
Zˆ R
Rot 2
Xˆ B
3
Zˆ R
 Xˆ B  cos 2 0 - sinθ2   Xˆ R 
 ˆ  
 ˆ 

1
0   YR 
 YB    0
 Zˆ   sinθ
  Zˆ R 
0
cos
θ
B
2
2

    
Xˆ R
 Xˆ B   cos 3 sinθ3 0  Xˆ R 
 ˆ  
 ˆ 

 YB   - sinθ3 cosθ3 0  YR 
 Zˆ   0
  Zˆ R 
0
1
B

    
Rot 3
Transformation from ECI to PQW
First do a three axis rotation of W, then a one axis rotation of I, then a three axis
rotation of w:
0
0   cosW sinW 0  ri 
rp   cosw sinw 0 1
 r   - sinw cosw 0 0 cosi sini  - sinW cosW 0 r 
 q 


 j 
rw   0
0
1 0 - sini cosi1   0
0
1 rk 
sin W cosw  cosW cosi sin w sin i sin w   ri 
 cosW cosw  sin W cosi sin w
  cosW sin w  sin W cosi cosw  sin W sin w  cosW cosi cosw sin i cosw  r j 

sin W sin i
 cosW sin i
cosi  rk 
wˆ
qˆ
pˆ
r(t) and v(t) in ECF
To get from PQW to ECI we invert the previous transformation, which
turns out to just be the transpose:
 ri  cosW cosw  sin W cosi sin w  cosW sin w  sin W cosi cosw sin W sin i  rp 
r   sin W cosw  cosW cosi sin w  sin W sin w  cosW cosi cosw  cosW sin i   r 
 j 
 q 
rk  
sin i sin w
sin i cosw
cosi  rw 
Greenwich meridian
To get from ECI to ECF we rotate
through the Greenwich mean
sidereal time:
r ECF  T r ECI , vECF  T vECI
GST
ECI
ECF
 cos(GST ) sin(GST) 0
T   sin(GST) cos(GST) 0

0
0
1
r(t) in SEZ
To get from ECF to the topocentric-horizon frame, SEZ, we rotate through latitude,
l, and longitude, f and subtract off the position vector to the site on the Earth:
r SEZ  T r ECF
 0 
  0 
 Rearth 
 sin f cosl sin f sin l  cosf 
T    sin l
cosl
0 
cosf cosl cosf sin l sin f 
SEZ
f
ECF
l
This vector can then be used to find the
azimuth and elevation of the satellite with
respect to the observer on the ground