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manual keplerian calculations




Yes, I'm that sick.

I'm working on solving the predicion problem with a pencil and paper. 
I've taken quite a bit of calculus, but that was quite some time ago.

Please reply directly.  I'll make sure your fine words get back to the 
list once the details are sorted out.  I'd appreciate direct email replies 
and I'll summarize to the list.

I have tracking software, I don't need to track via pencil and paper, I 
just want to know how to do it by hand.

I'm refering to the ARRL book AMatuer Satellite Handbook (Davidoff)

I'm starting with the Keplerian data for AO-7 from Keplerian Bulletin 78, 
ARLK078.

The epoch time is Oct 10, 2008 at 15:45:01.72Z

I attached the kep file for AO-7 for reference.

Any reference to values is about AO-7 from that Keplerian Update.

Here are my questions:


First to get the terms and understanding correct -- the Mean Anomoly is 
listed as 320.9603 deg.

Is this the value of E that is used on page 12-4 to 12-5, and I quote:

"In an eliptical orbit, time from perigree, t is given by:

t = ( T / 2pi) * (E - e *sin(E))     [ eq 12.10 ]

where the angle E, known as the eccentric anomaly, is defined by the 
associated equations:

E = 2 * arctan(  sqrt( (1-e)/(1+e)) * tan ( (theta)/2 ) + (360deg * n)

n = 0 when  -pi <= theta <= pi
or 1 when pi < theta <= (3pi)"


OK, so it says E (the eccentric anomaly) is a function of theta and 
eccentricity.

Eccentricy of the orbit is in the Keps.

Let's just assume I'm trying to compute the lat/long of AO-7 at the epoch.

Is the RAAN in the Keps the longitude of the satellite at the epoch?

Is the mean anomaly the longitude of the satellite at the epoch?

I need the value of E to compute the value of t for this reason:

(skip ahead to page 12-18 from the Handbook mentioned above)

Finding latitude of the satellite, phi, at perigree is:

phi(p) = arcsin(  sin(i) * sin(w) )

where i = inclination, w = angle between the ascending node and perigree.

Using equation 12.23, to find that latitude of the SSP:

phi(t) = arcsin (  sin(i) * sin( phi(t) + w) )   [Eq. 12.23 ]

Is that a misprint?  the value of phi(t) depends on phi(t) ?

The gap here I'm trying to understand is, computing t (the elapsed time 
since the ascending node) so I can plug it into the formula for latitude 
[ phi as a function of t, phi(t) ]


Summary of questions:

Which data from the kep do I use to compute t based on equation 12.10

What does RAAN and Mean Anomaly mean at the Epoch?

Is the bird at latitude zero at epoch?

Computing a from Keplers laws,  a = 31.25 * T^(2/3).

Is this T the Anomalistic Period?

Using that value of a, and eccentricity e, I can compute

w_dot = 4.97 * (Req/a)^(3.5) * (5 * cos^2(i) -1)/ ((1-e^2)^2)  [eq 12.13a]

How does w_dot relate to ang of apogee from the kep data?


Thanks,

-jeff

W7BRS

ps. Again, reply directly -- this sort of question can make enough spam on 
the DL to last you through Festivus.
SB KEP @ ARL $ARLK078
ARLK078 Keplerian data

ZCZC SK78
QST de W1AW  
Keplerian Bulletin 78  ARLK078
>From ARRL Headquarters
Newington, CT  October 10, 2008
To all radio amateurs 

SB KEP ARL ARLK078
ARLK078 Keplerian data

Special thanks to AMSAT-NA (AMSAT.ORG) for the following Keplerian
data.

Decode 2-line elsets with the following key:
1 AAAAAU 00  0  0 BBBBB.BBBBBBBB  .CCCCCCCC  00000-0  00000-0 0  DDDZ
2 AAAAA EEE.EEEE FFF.FFFF GGGGGGG HHH.HHHH III.IIII JJ.JJJJJJJJKKKKKZ
KEY: A-CATALOGNUM B-EPOCHTIME C-DECAY D-ELSETNUM E-INCLINATION F-RAAN
G-ECCENTRICITY H-ARGPERIGEE I-MNANOM J-MNMOTION K-ORBITNUM Z-CHECKSUM

AO-07
1 07530U 74089B   08283.65627308 -.00000027  00000-0  10000-3 0  2976
2 07530 101.4568 311.2861 0012054 039.2337 320.9603 12.53574392551323

Keplerian bulletins are transmitted twice weekly from W1AW.  The
next scheduled transmission of these data will be Tuesday, October 
14, 2008, at 2230z on Baudot and AMTOR.
NNNN
/EX

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