Satellite Position: Calculating ECI from ECEF

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Imagin_e
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Homework Statement


Hi!

I just found this forum after trying to calculate something regarding a satellite's orbit. Hope that you can help me here. So, I need to calculate the matrix of a satellite's site vectors. I have found it in the ECEF form, but since the satellite is doing observations in different times, I need it in ECI for this particular problem.

Homework Equations


See below.

The Attempt at a Solution


I searched and for the ECEF position, we have the following equations to use: (see the attached file)

I have, from my values and by using these equation, the site position in ECEF:

r = [-7166.056 -6364.272 -3210.412
-1309.400 -186.778 824.20
669.27 -3545.190 -6467.48 ]

How do I get the ECI position now? I guess that I need to consider the rotation of the Earth and the time between the observations. Is a matrix multiplication necessary?

Thanks in advance!
 
Attachments
  • Handwritten-style typeset equations for converting a site position from geodetic inputs to ECEF coordinates: C = R/sqrt(1 - e^2 sin^2(phi_gd)), S = C(1 - e^2), r_delta = (C + h_ellp) cos(phi_gd), r_K = (S + h_ellp) sin(phi_gd), and the ECEF vector [r_delta cos(lambda), r_delta sin(lambda), r_K].
    EFEC position.png
    8.5 KB · Views: 674

Discussion

Welcome to PF! Your problem is very special. Yes, change of coordinate system involves matrix multiplication. You can ask about ECI and ECEF in general in the Forum "Classical Physics". If you want to ask about a homework problem, write the question in detail, and post it to Homework/Advanced Physics.
 
ehild said:
Welcome to PF! Your problem is very special. Yes, change of coordinate system involves matrix multiplication. You can ask about ECI and ECEF in general in the Forum "Classical Physics". If you want to ask about a homework problem, write the question in detail, and post it to Homework/Advanced Physics.
Thanks!
The thing is, I cannot find it here. Let be show what I've done so far. I have the data below, and I'm only going to look at the nr 3, 5 and 6 data (See data).

I calculated the Julian date (easy), which is:

JD1= 2,456,159.986 435
JD2= 2,456,159.991 991
JD3= 2,456,159.994

Next, the change in times between the observations : tau1= 2,456,159.986 435− 2,456,159.991 991 = -8 min and
tau2=2,456,159.994 769 − 2,456,159.991 991 = 4 min

The I used the equations I posted above and got (see EFEC)

NOW I'm stuck, how should I rotate/transform it to get the ECI position?
 
Attachments
  • Table of six observations with columns Obs. #, time (h min s), alpha t in degrees and delta t in degrees; times run 11:32:28 to 11:48:28 at 4-minute intervals.
    data.png
    4.8 KB · Views: 570
  • Column vector r_site_ECEF equal to [-1673.9286, -4599.0809, 4079.2711] km, with components listed top to bottom.
    ECEF.png
    1.9 KB · Views: 599
Also, I know how the formula for ECEF to ECI looks like (rather a matrix), but how should I perform the operation?
 
Attachments
  • Equation C_i^e = R_3(omega_e t) equals a 3x3 rotation matrix with rows (cos omega_e t, sin omega_e t, 0), (-sin omega_e t, cos omega_e t, 0), and (0, 0, 1).
    Screen Shot 2015-10-19 at 12.07.29.png
    2.3 KB · Views: 590
Soory, I am not familiar with the topic. I ask the Mentors to move your thread to Advanced Physics.
 

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