MAE 5410

1
MAE 5410 AstrodynamicsLecture 5

• Orbit in Space
• Coordinate Frames and Time

2
Orienting the orbit plane
So far, weve 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 well 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 Earths 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, well switch to J2050.
3
Inclination, i
Angle between the orbit plane and the equatorial
plane
Increasing the orbital inclination increases the
maximum latitude of the groundtrack (in fact, the
maximum latitude equals the orbit inclination)
4
Longitude of the Ascending Node, W
Angle between the X-axis and the intersection of
the orbit plane and equatorial plane (the nodal
vector)
5
Argument of Perigee, w
Angle from the nodal vector to the periapsis
point (eccentricity vector, or )
6
Putting it all together
7
Some special cases
8
r(t) and v(t) in ECI
In Lecture 3 we found the position and velocity
in the PQW frame
In this lecture we defined orbital elements that
locate the PQW frame wrt the ECI frame.
To get from PQW to ECI, we perform a coordinate
transformation
9
Single Axis Rotations
10
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
11
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
To get from ECI to ECF we rotate through the
Greenwich mean sidereal time
Greenwich meridian
ECF
GST
ECI
12
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
This vector can then be used to find the azimuth
and elevation of the satellite with respect to
the observer on the ground
SEZ
ECF
f
l