THE SGP MODEL

The NORAD mean element sets can be used for prediction with SGP. All symbols not defined below are defined in the list of symbols in Section Twelve. Predictions are made by first calculating the constants

#math10#

a₁ = #tex2html_wrap_indisplay3558##tex2html_wrap_indisplay3559##tex2html_wrap_indisplay3560#

#math11#

δ₁ = #tex2html_wrap_indisplay3562#J₂#tex2html_wrap_indisplay3563##tex2html_wrap_indisplay3564#

#math12#

a_o = a₁#tex2html_wrap_indisplay3566#1 - #tex2html_wrap_indisplay3567#δ₁ - δ₁² - #tex2html_wrap_indisplay3570#δ₁³#tex2html_wrap_indisplay3573#

#math13#

p_o = a_o(1 - e_o²)

#math14#

q_o = a_o(1 - e_o)

#math15#

L_o = M_o + ω_o + Ω_o

#math16#

#tex2html_wrap_indisplay3580# = - #tex2html_wrap_indisplay3581#J₂#tex2html_wrap_indisplay3582#n_ocos i_o

#math17#

#tex2html_wrap_indisplay3584# = #tex2html_wrap_indisplay3585#J₂#tex2html_wrap_indisplay3586#n_o(5 cos²i_o - 1).

The secular effects of atmospheric drag and gravitation are included through the equations

#math18#

a = a_o#tex2html_wrap_indisplay3588##tex2html_wrap_indisplay3589##tex2html_wrap_indisplay3590#

#math19#

e = #tex2html_wrap_indisplay3592##tex2html_wrap_indisplay3593##tex2html_wrap_indisplay3594#

#math20#

p = a(1 - e²)

#math21#

Ω_{s_o} = Ω_o + #tex2html_wrap_indisplay3597#(t - t_o)

#math22#

ω_{s_o} = ω_o + #tex2html_wrap_indisplay3599#(t - t_o)

#math23#

L_s = L_o + #tex2html_wrap_indisplay3601#n_o + #tex2html_wrap_indisplay3602# + #tex2html_wrap_indisplay3603##tex2html_wrap_indisplay3604#(t - t_o) + #tex2html_wrap_indisplay3605#(t - t_o)² + #tex2html_wrap_indisplay3606#(t - t_o)³

where (t - t_o) is time since epoch.

Long-period periodics are included through the equations

#math24#

a_yNSL = e sinω_{s_o} - #tex2html_wrap_indisplay3609##tex2html_wrap_indisplay3610##tex2html_wrap_indisplay3611#sin i_o

#math25#

L = L_s - #tex2html_wrap_indisplay3613##tex2html_wrap_indisplay3614##tex2html_wrap_indisplay3615#a_xNSLsin i_o#tex2html_wrap_indisplay3616##tex2html_wrap_indisplay3617##tex2html_wrap_indisplay3618#

where

#math26#

a_xNSL = e cosω_{s_o}.

Solve Kepler's equation for E + ω (by iteration to the desired accuracy), where

#math27#

(E + ω)_i+1 = (E + ω)_i + Δ(E + ω)_i

with

#math28#

Δ(E + ω)_i = #tex2html_wrap_indisplay3623#

#math29#

U = L - Ω_{s_o}

and

#math30#

(E + ω)₁ = U.

Then calculate the intermediate (partially osculating) quantities

#math31#

e cos E = a_xNSLcos(E + ω) + a_yNSLsin(E + ω)

#math32#

e sin E = a_xNSLsin(E + ω) - a_yNSLcos(E + ω)

#math33#

e_L² = (a_xNSL)² + (a_yNSL)²

#math34#

p_L = a(1 - e_L²)

#math35#

r = a(1 - e cos E)

#math36#

#tex2html_wrap_indisplay3636# = k_e#tex2html_wrap_indisplay3637#e sin E

#math37#

r#tex2html_wrap_indisplay3639# = k_e#tex2html_wrap_indisplay3640#

#math38#

sin u = #tex2html_wrap_indisplay3642##tex2html_wrap_indisplay3643#sin(E + ω) - a_yNSL - a_xNSL#tex2html_wrap_indisplay3644##tex2html_wrap_indisplay3645#

#math39#

cos u = #tex2html_wrap_indisplay3647##tex2html_wrap_indisplay3648#cos(E + ω) - a_xNSL + a_yNSL#tex2html_wrap_indisplay3649##tex2html_wrap_indisplay3650#

#math40#

u = tan^-1#tex2html_wrap_indisplay3652##tex2html_wrap_indisplay3653##tex2html_wrap_indisplay3654#.

Short-period perturbations are now included by

#math41#

r_k = r + #tex2html_wrap_indisplay3656#J₂#tex2html_wrap_indisplay3657#sin²i_ocos 2u

#math42#

u_k = u - #tex2html_wrap_indisplay3659#J₂#tex2html_wrap_indisplay3660#(7 cos²i_o -1)sin 2u

#math43#

Ω_k = Ω_{s_o} + #tex2html_wrap_indisplay3662#J₂#tex2html_wrap_indisplay3663#cos i_osin 2u

#math44#

i_k = i_o + #tex2html_wrap_indisplay3665#J₂#tex2html_wrap_indisplay3666#sin i_ocos i_ocos 2u.

Then unit orientation vectors are calculated by

#math45#

#tex2html_wrap_indisplay3668# = #tex2html_wrap_indisplay3669#sin u_k + #tex2html_wrap_indisplay3670#cos u_k

#math46#

#tex2html_wrap_indisplay3672# = #tex2html_wrap_indisplay3673#cos u_k - #tex2html_wrap_indisplay3674#sin u_k

where

#math47#

#tex2html_wrap_indisplay3676# = #tex2html_wrap_indisplay3677##tex2html_wrap_indisplay3678##tex2html_wrap_indisplay3679#

#math48#

#tex2html_wrap_indisplay3681# = #tex2html_wrap_indisplay3682##tex2html_wrap_indisplay3683##tex2html_wrap_indisplay3684#.

Then position and velocity are given by

#math49#

#tex2html_wrap_indisplay3686# = r_k#tex2html_wrap_indisplay3687#

and

#math50#

#tex2html_wrap_indisplay3689# = #tex2html_wrap_indisplay3690##tex2html_wrap_indisplay3691# + (r#tex2html_wrap_indisplay3692#)#tex2html_wrap_indisplay3693#.

A FORTRAN IV computer code listing of the subroutine SGP is given below. #center3694#