MAE 5410 – AstrodynamicsLecture 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

Putting it all together

Some special cases

l uf

f

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