THE SGP8 MODEL

The NORAD mean element sets can be used for prediction with SGP8. 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

#math194#

a₁ = #tex2html_wrap_indisplay4159##tex2html_wrap_indisplay4160##tex2html_wrap_indisplay4161#

#math195#

δ₁ = #tex2html_wrap_indisplay4163##tex2html_wrap_indisplay4164##tex2html_wrap_indisplay4165#

#math196#

a_o = a₁#tex2html_wrap_indisplay4167#1 - #tex2html_wrap_indisplay4168#δ₁ - δ₁² - #tex2html_wrap_indisplay4171#δ₁³#tex2html_wrap_indisplay4174#

#math197#

δ_o = #tex2html_wrap_indisplay4176##tex2html_wrap_indisplay4177##tex2html_wrap_indisplay4178#

#math198#

n''_o = #tex2html_wrap_indisplay4180#

#math199#

a''_o = #tex2html_wrap_indisplay4182#.

The ballistic coefficient (B term) is then calculated from the B^* drag term by

#math200#

B = 2B^*/ρ_o

where

#math201#

ρ_o = (2.461×10^-5)#tex2html_wrap_indisplay4187#

is a reference value of atmospheric density.

Then calculate the constants

#math202#

β² = 1 - e²

#math203#

θ = cos i

#math204#

#tex2html_wrap_indisplay4191# = - #tex2html_wrap_indisplay4192##tex2html_wrap_indisplay4193#(1 - 3θ²)

#math205#

#tex2html_wrap_indisplay4195# = - #tex2html_wrap_indisplay4196##tex2html_wrap_indisplay4197#(1 - 5θ²)

#math206#

#tex2html_wrap_indisplay4199# = - 3#tex2html_wrap_indisplay4200#θ

#math207#

#tex2html_wrap_indisplay4202# = #tex2html_wrap_indisplay4203##tex2html_wrap_indisplay4204#(13 - 78θ² +137θ⁴)

#math208#

#tex2html_wrap_indisplay4206# = #tex2html_wrap_indisplay4207##tex2html_wrap_indisplay4208#(7 - 114θ² +395θ⁴) + #tex2html_wrap_indisplay4209##tex2html_wrap_indisplay4210#(3 - 36θ² +49θ⁴)

#math209#

#tex2html_wrap_indisplay4212# = #tex2html_wrap_indisplay4213##tex2html_wrap_indisplay4214#θ(4 - 19θ²) + #tex2html_wrap_indisplay4215##tex2html_wrap_indisplay4216#θ(3 - 7θ²)

#math210#

#tex2html_wrap_indisplay4218# = n'' + #tex2html_wrap_indisplay4219# + #tex2html_wrap_indisplay4220#

#math211#

#tex2html_wrap_indisplay4222# = #tex2html_wrap_indisplay4223# + #tex2html_wrap_indisplay4224#

#math212#

#tex2html_wrap_indisplay4226# = #tex2html_wrap_indisplay4227# + #tex2html_wrap_indisplay4228#

#math213#

ξ = #tex2html_wrap_indisplay4230#

#math214#

η = esξ

#math215#

ψ = #tex2html_wrap_indisplay4233#

#math216#

α² = 1 + e²

#math217#

C_o = #tex2html_wrap_indisplay4236#Bρ_o(q_o - s)⁴n''a''ξ⁴α^-1ψ^-7

#math218#

C₁ = #tex2html_wrap_indisplay4238#n''α⁴C_o

#math219#

D₁ = ξψ^-2/a''β²

#math220#

D₂ = 12 + 36η² + #tex2html_wrap_indisplay4241#η⁴

#math221#

D₃ = 15η² + #tex2html_wrap_indisplay4243#η⁴

#math222#

D₄ = 5η + #tex2html_wrap_indisplay4245#η³

#math223#

D₅ = ξψ^-2

#math224#

B₁ = - k₂(1 - 3θ²)

#math225#

B₂ = - k₂(1 - θ²)

#math226#

B₃ = #tex2html_wrap_indisplay4250#sin i

#math227#

C₂ = D₁D₃B₂

#math228#

C₃ = D₄D₅B₃

#math229#

#tex2html_wrap_indisplay4254# = C₁#tex2html_wrap_indisplay4255#2 + 3η² +20eη +5eη³ + #tex2html_wrap_indisplay4256#e² +34e²η² + D₁D₂B₁ + C₂cos 2ω + C₃sinω#tex2html_wrap_indisplay4257#

#math230#

C₄ = D₁D₇B₂

#math231#

C₅ = D₅D₈B₃

#math232#

D₆ = 30η + #tex2html_wrap_indisplay4261#η³

#math233#

D₇ = 5η + #tex2html_wrap_indisplay4263#η³

#math234#

D₈ = 1 + #tex2html_wrap_indisplay4265#η² + η⁴

#math235#

#tex2html_wrap_indisplay4267# = - C_o#tex2html_wrap_indisplay4268#4η + η³ +5e + 15eη² + #tex2html_wrap_indisplay4269#e²η +7e²η³ + D₁D₆B₁ + C₄cos 2ω + C₅sinω#tex2html_wrap_indisplay4270#

#math236#

#tex2html_wrap_indisplay4272#/α = e#tex2html_wrap_indisplay4273#α^-2

#math237#

C₆ = #tex2html_wrap_indisplay4275##tex2html_wrap_indisplay4276#

#math238#

#tex2html_wrap_indisplay4278#/ξ = 2a''ξ(C₆β² + e#tex2html_wrap_indisplay4279#)

#math239#

#tex2html_wrap_indisplay4281# = (#tex2html_wrap_indisplay4282# + e#tex2html_wrap_indisplay4283#/ξ)sξ

#math240#

#tex2html_wrap_indisplay4285#/ψ = - η#tex2html_wrap_indisplay4286#ψ^-2

#math241#

#tex2html_wrap_indisplay4288#/C_o = C₆ +4#tex2html_wrap_indisplay4289#/ξ - #tex2html_wrap_indisplay4290#/α -7#tex2html_wrap_indisplay4291#/ψ

#math242#

#tex2html_wrap_indisplay4293#/C₁ = #tex2html_wrap_indisplay4294#/n'' + 4#tex2html_wrap_indisplay4295#/α + #tex2html_wrap_indisplay4296#/C_o

#math243#

D₉ = 6η +20e + 15eη² +68e²η

#math244#

D₁₀ = 20η +5η³ +17e + 68eη²

#math245#

D₁₁ = 72η +18η³

#math246#

D₁₂ = 30η +10η³

#math247#

D₁₃ = 5 + #tex2html_wrap_indisplay4302#η²

#math248#

D₁₄ = #tex2html_wrap_indisplay4304#/ξ -2#tex2html_wrap_indisplay4305#/ψ

#math249#

D₁₅ = 2(C₆ + e#tex2html_wrap_indisplay4307#β^-2)

#math250#

#tex2html_wrap_indisplay4309# = D₁(D₁₄ + D₁₅)

#math251#

#tex2html_wrap_indisplay4311# = #tex2html_wrap_indisplay4312#D₁₁

#math252#

#tex2html_wrap_indisplay4314# = #tex2html_wrap_indisplay4315#D₁₂

#math253#

#tex2html_wrap_indisplay4317# = #tex2html_wrap_indisplay4318#D₁₃

#math254#

#tex2html_wrap_indisplay4320# = D₅D₁₄

#math255#

#tex2html_wrap_indisplay4322# = B₂(#tex2html_wrap_indisplay4323#D₃ + D₁#tex2html_wrap_indisplay4324#)

#math256#

#tex2html_wrap_indisplay4326# = B₃(#tex2html_wrap_indisplay4327#D₄ + D₅#tex2html_wrap_indisplay4328#)

#math257#

#tex2html_wrap_indisplay4330# = - #tex2html_wrap_indisplay4331##tex2html_wrap_indisplay4332#(1 - 5θ²)

#math258#

D₁₆ = D₉#tex2html_wrap_indisplay4334# + D₁₀#tex2html_wrap_indisplay4335# + B₁(#tex2html_wrap_indisplay4336#D₂ + D₁#tex2html_wrap_indisplay4337#) + #tex2html_wrap_indisplay4338#cos 2ω + #tex2html_wrap_indisplay4339#sinω + #tex2html_wrap_indisplay4340#(C₃cosω -2C₂sin 2ω)

#math259#

#tex2html_wrap_indisplay4342# = #tex2html_wrap_indisplay4343##tex2html_wrap_indisplay4344#/C₁ + C₁D₁₆

#math260#

#tex2html_wrap_indisplay4346# = #tex2html_wrap_indisplay4347#

#math261#

D₁₇ = #tex2html_wrap_indisplay4349#/n'' - (#tex2html_wrap_indisplay4350#/n'')²

#math262#

#tex2html_wrap_indisplay4352#/ξ = 2(#tex2html_wrap_indisplay4353#/ξ - C₆)#tex2html_wrap_indisplay4354#/ξ +2a''ξ#tex2html_wrap_indisplay4355##tex2html_wrap_indisplay4356#D₁₇β² -2C₆e#tex2html_wrap_indisplay4357# + #tex2html_wrap_indisplay4358# + e#tex2html_wrap_indisplay4359##tex2html_wrap_indisplay4360#

#math263#

#tex2html_wrap_indisplay4362# = (#tex2html_wrap_indisplay4363# +2#tex2html_wrap_indisplay4364##tex2html_wrap_indisplay4365#/ξ)sξ + η#tex2html_wrap_indisplay4366#/ξ

#math264#

D₁₈ = #tex2html_wrap_indisplay4368#/ξ - (#tex2html_wrap_indisplay4369#/ξ)²

#math265#

D₁₉ = - (#tex2html_wrap_indisplay4371#/ψ)²(1 + η^-2) - η#tex2html_wrap_indisplay4372#ψ^-2

#math266#

#tex2html_wrap_indisplay4374# = #tex2html_wrap_indisplay4375#(D₁₄ + D₁₅) + D₁#tex2html_wrap_indisplay4376#D₁₈ -2D₁₉ + #tex2html_wrap_indisplay4377#D₁₇ +2α²#tex2html_wrap_indisplay4378#β^-4 +2e#tex2html_wrap_indisplay4379#β^-2#tex2html_wrap_indisplay4380#

#math267#

#tex2html_wrap_indisplay4382#_o = #tex2html_wrap_indisplay4383#

#math268#

p = #tex2html_wrap_indisplay4385#

#math269#

γ = - #tex2html_wrap_indisplay4387##tex2html_wrap_indisplay4388#

#math270#

n_D = #tex2html_wrap_indisplay4390#

#math271#

q = 1 - #tex2html_wrap_indisplay4392#

#math272#

e_D = #tex2html_wrap_indisplay4394#

where all quantities are epoch values.

The secular effects of atmospheric drag and gravitation are included by

#math273#

n = n''_o + n_D[1 - (1 - γ(t - t_o))^p]

#math274#

e = e_o + e_D[1 - (1 - γ(t - t_o))^q]

#math275#

ω = ω_o + #tex2html_wrap_indisplay4398##tex2html_wrap_indisplay4399#(t - t_o) + #tex2html_wrap_indisplay4400##tex2html_wrap_indisplay4401#Z₁#tex2html_wrap_indisplay4402# + #tex2html_wrap_indisplay4403#(t - t_o)

#math276#

Ω = Ω_o'' + #tex2html_wrap_indisplay4405##tex2html_wrap_indisplay4406#(t - t_o) + #tex2html_wrap_indisplay4407##tex2html_wrap_indisplay4408#Z₁#tex2html_wrap_indisplay4409# + #tex2html_wrap_indisplay4410#(t - t_o)

#math277#

M = M_o + n''_o(t - t_o) + Z₁ + #tex2html_wrap_indisplay4412##tex2html_wrap_indisplay4413#(t - t_o) + #tex2html_wrap_indisplay4414##tex2html_wrap_indisplay4415#Z₁#tex2html_wrap_indisplay4416# + #tex2html_wrap_indisplay4417#(t - t_o)

where

#math278#

Z₁ = #tex2html_wrap_indisplay4419##tex2html_wrap_indisplay4420#(t - t_o) + #tex2html_wrap_indisplay4421#[(1 - γ(t - t_o))^p+1 - 1]#tex2html_wrap_indisplay4422#.

If drag is very small (#math279##tex2html_wrap_inline4424# less than #math280#1.5×10^-6/min) then the secular equations for n, e, and Z₁ should be replaced by

#math281#

n = n''_o + #tex2html_wrap_indisplay4430#(t - t_o)

#math282#

e = e''_o + #tex2html_wrap_indisplay4432#(t - t_o)

#math283#

Z₁ = #tex2html_wrap_indisplay4434##tex2html_wrap_indisplay4435#(t - t_o)²

where (t - t_o) is time since epoch and where

#math284#

#tex2html_wrap_indisplay4438# = - #tex2html_wrap_indisplay4439##tex2html_wrap_indisplay4440#(1 - e_o).

Solve Kepler's equation for E by using the iteration equation

#math285#

E_i+1 = E_i + ΔE_i

with

#math286#

ΔE_i = #tex2html_wrap_indisplay4444#

and

#math287#

E₁ = M + e sin M + #tex2html_wrap_indisplay4446#e²sin 2M.

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

#math288#

a = #tex2html_wrap_indisplay4448##tex2html_wrap_indisplay4449##tex2html_wrap_indisplay4450#

#math289#

β = (1 - e²)^{#tex2html_wrap_indisplay4452#}

#math290#

sin f = #tex2html_wrap_indisplay4454#

#math291#

cos f = #tex2html_wrap_indisplay4456#

#math292#

u = f + ω

#math293#

r'' = #tex2html_wrap_indisplay4459#

#math294#

#tex2html_wrap_indisplay4461# = #tex2html_wrap_indisplay4462#sin f

#math295#

(r#tex2html_wrap_indisplay4464#)'' = #tex2html_wrap_indisplay4465#

#math296#

δr = #tex2html_wrap_indisplay4467##tex2html_wrap_indisplay4468#[(1 - θ²)cos 2u + 3(1 - 3θ²)] - #tex2html_wrap_indisplay4469##tex2html_wrap_indisplay4470#sin i_osin u

#math297#

δ#tex2html_wrap_indisplay4472# = - n#tex2html_wrap_indisplay4473##tex2html_wrap_indisplay4474##tex2html_wrap_indisplay4475##tex2html_wrap_indisplay4476##tex2html_wrap_indisplay4477#(1 - θ²)sin 2u + #tex2html_wrap_indisplay4478##tex2html_wrap_indisplay4479#sin i_ocos u#tex2html_wrap_indisplay4480#

#math298#

δI = θ#tex2html_wrap_indisplay4482##tex2html_wrap_indisplay4483##tex2html_wrap_indisplay4484#sin i_ocos 2u - #tex2html_wrap_indisplay4485##tex2html_wrap_indisplay4486#e sinω#tex2html_wrap_indisplay4487#

#math299#

δ(r#tex2html_wrap_indisplay4489#) = - n#tex2html_wrap_indisplay4490##tex2html_wrap_indisplay4491##tex2html_wrap_indisplay4492#δr + na#tex2html_wrap_indisplay4493##tex2html_wrap_indisplay4494##tex2html_wrap_indisplay4495##tex2html_wrap_indisplay4496#δI

#math300#

δu = #tex2html_wrap_indisplay4498#

#math301#

δλ = #tex2html_wrap_indisplay4500#

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

#math302#

r = r'' + δr

#math303#

#tex2html_wrap_indisplay4503# = #tex2html_wrap_indisplay4504# + δ#tex2html_wrap_indisplay4505#

#math304#

r#tex2html_wrap_indisplay4507# = (r#tex2html_wrap_indisplay4508#)'' + δ(r#tex2html_wrap_indisplay4509#)

#math305#

y₄ = sin#tex2html_wrap_indisplay4511#sin u + cos u sin#tex2html_wrap_indisplay4512#δu + #tex2html_wrap_indisplay4513#sin u cos#tex2html_wrap_indisplay4514#δI

#math306#

y₅ = sin#tex2html_wrap_indisplay4516#cos u - sin u sin#tex2html_wrap_indisplay4517#δu + #tex2html_wrap_indisplay4518#cos u cos#tex2html_wrap_indisplay4519#δI

#math307#

λ = u + Ω + δλ.

Unit orientation vectors are calculated by

#math308#

U_x = 2y₄(y₅sinλ - y₄cosλ) + cosλ

#math309#

U_y = - 2y₄(y₅cosλ + y₄sinλ) + sinλ

#math310#

U_z = 2y₄cos#tex2html_wrap_indisplay4524#

#math311#

V_x = 2y₅(y₅sinλ - y₄cosλ) - sinλ

#math312#

V_y = - 2y₅(y₅cosλ + y₄sinλ) + cosλ

#math313#

V_z = 2y₅cos#tex2html_wrap_indisplay4528#

where

#math314#

cos#tex2html_wrap_indisplay4530# = #tex2html_wrap_indisplay4531#.

Position and velocity are given by

#math315#

#tex2html_wrap_indisplay4533# = r#tex2html_wrap_indisplay4534#

#math316#

#tex2html_wrap_indisplay4536# = #tex2html_wrap_indisplay4537##tex2html_wrap_indisplay4538# + r#tex2html_wrap_indisplay4539##tex2html_wrap_indisplay4540#.

A FORTRAN IV computer code listing of the subroutine SGP8 is given below. #center4541#