THE SGP4 MODEL

The NORAD mean element sets can be used for prediction with SGP4. All symbols not defined below are defined in the list of symbols in Section Twelve. The original mean motion (n''o) and semimajor axis (a''o) are first recovered from the input elements by the equations

#math51#

a1 = #tex2html_wrap_indisplay3702##tex2html_wrap_indisplay3703##tex2html_wrap_indisplay3704#

#math52#

δ1 = #tex2html_wrap_indisplay3706##tex2html_wrap_indisplay3707##tex2html_wrap_indisplay3708#

#math53#

ao = a1#tex2html_wrap_indisplay3710#1 - #tex2html_wrap_indisplay3711#δ1 - δ12 - #tex2html_wrap_indisplay3714#δ13#tex2html_wrap_indisplay3717#

#math54#

δo = #tex2html_wrap_indisplay3719##tex2html_wrap_indisplay3720##tex2html_wrap_indisplay3721#

#math55#

n''o = #tex2html_wrap_indisplay3723#

#math56#

a''o = #tex2html_wrap_indisplay3725#.

For perigee between 98 kilometers and 156 kilometers, the value of the constant s used in SGP4 is changed to

#math57#

s* = a''o(1 - eo) - s + aE

For perigee below 98 kilometers, the value of s is changed to

#math58#

s* = 20/<#1#>XKMPER<#1#> + aE.

If the value of s is changed, then the value of (qo - s)4 must be replaced by

#math59#

(qo - s*)4 = #tex2html_wrap_indisplay3733#[(qo - s)4]#tex2html_wrap_indisplay3734# + s - s*#tex2html_wrap_indisplay3735#.

Then calculate the constants (using the appropriate values of s and (qo - s)4)

#math60#

θ = cos io

#math61#

ξ = #tex2html_wrap_indisplay3740#

#math62#

βo = (1 - eo2)#tex2html_wrap_indisplay3744#

#math63#

η = a''oeoξ

#math64#

C2 = #tex2html_wrap_indisplay3747#

#math65#

C1 = B*C2

#math66#

C3 = #tex2html_wrap_indisplay3750#

#math67#

C4 = #tex2html_wrap_indisplay3752#

#math68#

C5 = 2(qo - s)4ξ4a''oβo2(1 - η2)-#tex2html_wrap_indisplay3756##tex2html_wrap_indisplay3757#1 + #tex2html_wrap_indisplay3758#η(η + eo) + eoη3#tex2html_wrap_indisplay3759#

#math69#

D2 = 4a''oξC12

#math70#

D3 = #tex2html_wrap_indisplay3764#a''oξ2(17a''o + s)C13

#math71#

D4 = #tex2html_wrap_indisplay3768#a''oξ3(221a''o +31s)C14.

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

#math72#

MDF = Mo + #tex2html_wrap_indisplay3772#1 + #tex2html_wrap_indisplay3773# + #tex2html_wrap_indisplay3774##tex2html_wrap_indisplay3775#n''o(t - to)

#math73#

ωDF = #tex2html_wrap_indisplay3777#

#math74#

ΩDF = Ωo + #tex2html_wrap_indisplay3779# - #tex2html_wrap_indisplay3780# + #tex2html_wrap_indisplay3781# + #tex2html_wrap_indisplay3782##tex2html_wrap_indisplay3783#n''o(t - to)

#math75#

δω = B*C3(cosωo)(t - to)

#math76#

δM = - #tex2html_wrap_indisplay3786#(qo - s)4B*ξ4#tex2html_wrap_indisplay3787#[(1 + ηcos MDF)3 - (1 + ηcos Mo)3]

#math77#

Mp = MDF + δω + δM

#math78#

ω = ωDF - δω - δM

#math79#

Ω = ΩDF - #tex2html_wrap_indisplay3791##tex2html_wrap_indisplay3792#C1(t - to)2

#math80#

e = eo - B*C4(t - to) - B*C5(sin Mp - sin Mo)

#math81#

a = a''o[1 - C1(t - to) - D2(t - to)2 - D3(t - to)3 - D4(t - to)4]2

#math82#

I   L = #tex2html_wrap_indisplay3796#

#math83#

β = #tex2html_wrap_indisplay3798#

#math84#

n = ke#tex2html_wrap_indisplay3800#a#tex2html_wrap_indisplay3801#

where (t - to) is time since epoch. It should be noted that when epoch perigee height is less than 220 kilometers, the equations for a and #math85#I   L are truncated after the C1 term, and the terms involving C5, #math86#δω, and δM are dropped.

Add the long-period periodic terms

#math87#

axN = e cosω

#math88#

I   LL = #tex2html_wrap_indisplay3813#(e cosω)#tex2html_wrap_indisplay3814##tex2html_wrap_indisplay3815##tex2html_wrap_indisplay3816#

#math89#

ayNL = #tex2html_wrap_indisplay3818#

#math90#

I   LT = I   L + I   LL

#math91#

ayN = e sinω + ayNL.

Solve Kepler's equation for #math92#(E + ω) by defining

#math93#

U = I   LT - Ω

and using the iteration equation

#math94#

(E + ω)i+1 = (E + ω)i + Δ(E + ω)i

with

#math95#

Δ(E + ω)i = #tex2html_wrap_indisplay3831#

and

#math96#

(E + ω)1 = U.

The following equations are used to calculate preliminary quantities needed for short-period periodics.

#math97#

e cos E = axNcos(E + ω) + ayNsin(E + ω)

#math98#

e sin E = axNsin(E + ω) - ayNcos(E + ω)

#math99#

eL = (axN2 + ayN2)#tex2html_wrap_indisplay3840#

#math100#

pL = a(1 - eL2)

#math101#

r = a(1 - e cos E)

#math102#

#tex2html_wrap_indisplay3846# = ke#tex2html_wrap_indisplay3847#e sin E

#math103#

r#tex2html_wrap_indisplay3849# = ke#tex2html_wrap_indisplay3850#

#math104#

cos u = #tex2html_wrap_indisplay3852##tex2html_wrap_indisplay3853#cos(E + ω) - axN + #tex2html_wrap_indisplay3854##tex2html_wrap_indisplay3855#

#math105#

sin u = #tex2html_wrap_indisplay3857##tex2html_wrap_indisplay3858#sin(E + ω) - ayN - #tex2html_wrap_indisplay3859##tex2html_wrap_indisplay3860#

#math106#

u = tan-1#tex2html_wrap_indisplay3862##tex2html_wrap_indisplay3863##tex2html_wrap_indisplay3864#

#math107#

Δr = #tex2html_wrap_indisplay3866#(1 - θ2)cos 2u

#math108#

Δu = - #tex2html_wrap_indisplay3868#(7θ2 -1)sin 2u

#math109#

ΔΩ = #tex2html_wrap_indisplay3870#sin 2u

#math110#

Δi = #tex2html_wrap_indisplay3872#sin iocos 2u

#math111#

Δ#tex2html_wrap_indisplay3874# = - #tex2html_wrap_indisplay3875#(1 - θ2)sin 2u

#math112#

Δr#tex2html_wrap_indisplay3877# = #tex2html_wrap_indisplay3878##tex2html_wrap_indisplay3879#(1 - θ2)cos 2u - #tex2html_wrap_indisplay3880#(1 - 3θ2)#tex2html_wrap_indisplay3881#

The short-period periodics are added to give the osculating quantities

#math113#

rk = r#tex2html_wrap_indisplay3883#1 - #tex2html_wrap_indisplay3884#k2#tex2html_wrap_indisplay3885#(3θ2 - 1)#tex2html_wrap_indisplay3886# + Δr

#math114#

uk = u + Δu

#math115#

Ωk = Ω + ΔΩ

#math116#

ik = io + Δi

#math117#

#tex2html_wrap_indisplay3891# = #tex2html_wrap_indisplay3892# + Δ#tex2html_wrap_indisplay3893#

#math118#

r#tex2html_wrap_indisplay3895# = r#tex2html_wrap_indisplay3896# + Δr#tex2html_wrap_indisplay3897#.

Then unit orientation vectors are calculated by

#math119#

#tex2html_wrap_indisplay3899# = #tex2html_wrap_indisplay3900#sin uk + #tex2html_wrap_indisplay3901#cos uk

#math120#

#tex2html_wrap_indisplay3903# = #tex2html_wrap_indisplay3904#cos uk - #tex2html_wrap_indisplay3905#sin uk

where

#math121#

#tex2html_wrap_indisplay3907# = #tex2html_wrap_indisplay3908##tex2html_wrap_indisplay3909##tex2html_wrap_indisplay3910#

#math122#

#tex2html_wrap_indisplay3912# = #tex2html_wrap_indisplay3913##tex2html_wrap_indisplay3914##tex2html_wrap_indisplay3915#.

Then position and velocity are given by

#math123#

#tex2html_wrap_indisplay3917# = rk#tex2html_wrap_indisplay3918#

and

#math124#

#tex2html_wrap_indisplay3920# = #tex2html_wrap_indisplay3921##tex2html_wrap_indisplay3922# + (r#tex2html_wrap_indisplay3923#)k#tex2html_wrap_indisplay3924#.

A FORTRAN IV computer code listing of the subroutine SGP4 is given below. These equations contain all currently anticipated changes to the SCC operational program. These changes are scheduled for implementation in March, 1981. #center3925#