THE SDP4 MODEL

The NORAD mean element sets can be used for prediction with SDP4. 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

#math125#

a₁ = #tex2html_wrap_indisplay3931##tex2html_wrap_indisplay3932##tex2html_wrap_indisplay3933#

#math126#

δ₁ = #tex2html_wrap_indisplay3935##tex2html_wrap_indisplay3936##tex2html_wrap_indisplay3937#

#math127#

a_o = a₁#tex2html_wrap_indisplay3939#1 - #tex2html_wrap_indisplay3940#δ₁ - δ₁² - #tex2html_wrap_indisplay3943#δ₁³#tex2html_wrap_indisplay3946#

#math128#

δ_o = #tex2html_wrap_indisplay3948##tex2html_wrap_indisplay3949##tex2html_wrap_indisplay3950#

#math129#

n''_o = #tex2html_wrap_indisplay3952#

#math130#

a''_o = #tex2html_wrap_indisplay3954#.

For perigee between 98 kilometers and 156 kilometers, the value of the constant s used in SDP4 is changed to

#math131#

s^* = a''_o(1 - e_o) - s + a_E.

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

#math132#

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

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

#math133#

(q_o - s^*)⁴ = #tex2html_wrap_indisplay3962#[(q_o - s)⁴]^{#tex2html_wrap_indisplay3963#} + s - s^*#tex2html_wrap_indisplay3964#.

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

#math134#

θ = cos i_o

#math135#

ξ = #tex2html_wrap_indisplay3969#

#math136#

β_o = (1 - e_o²)^{#tex2html_wrap_indisplay3973#}

#math137#

η = a''_oe_oξ

#math138#

C₂ = #tex2html_wrap_indisplay3976#

#math139#

C₁ = B^*C₂

#math140#

C₄ = #tex2html_wrap_indisplay3979#

#math141#

#tex2html_wrap_indisplay3981# = #tex2html_wrap_indisplay3982#1 + #tex2html_wrap_indisplay3983# + #tex2html_wrap_indisplay3984##tex2html_wrap_indisplay3985#n''_o

#math142#

#tex2html_wrap_indisplay3987# = #tex2html_wrap_indisplay3988# - #tex2html_wrap_indisplay3989# + #tex2html_wrap_indisplay3990# + #tex2html_wrap_indisplay3991##tex2html_wrap_indisplay3992#n''_o

#math143#

#tex2html_wrap_indisplay3994# = - #tex2html_wrap_indisplay3995#n''_o

#math144#

#tex2html_wrap_indisplay3997# = #tex2html_wrap_indisplay3998# + #tex2html_wrap_indisplay3999##tex2html_wrap_indisplay4000# + #tex2html_wrap_indisplay4001##tex2html_wrap_indisplay4002#n''_o.

At this point SDP4 calls the initialization section of DEEP which calculates all initialized quantities needed for the deep-space perturbations (see Section Ten).

The secular effects of gravity are included by

#math145#

M_DF = M_o + #tex2html_wrap_indisplay4004#(t - t_o)

#math146#

ω_DF = ω_o + #tex2html_wrap_indisplay4006#(t - t_o)

#math147#

Ω_DF = Ω_o + #tex2html_wrap_indisplay4008#(t - t_o)

where (t - t_o) is time since epoch. The secular effect of drag on longitude of ascending node is included by

#math148#

Ω = Ω_DF - #tex2html_wrap_indisplay4011##tex2html_wrap_indisplay4012#C₁(t - t_o)².

Next, SDP4 calls the secular section of DEEP which adds the deep-space secular effects and long-period resonance effects to the six classical orbital elements (see Section Ten).

The secular effects of drag are included in the remaining elements by

#math149#

a = a_DS[1 - C₁(t - t_o)]²

#math150#

e = e_DS - B^*C₄(t - t_o)

#math151#

I L = M_DS + ω_DS + Ω_DS + n''_o#tex2html_wrap_indisplay4016##tex2html_wrap_indisplay4017#C₁(t - t_o)²#tex2html_wrap_indisplay4018#

where a_DS, e_DS, M_DS, #math152#ω_DS, and #math153#Ω_DS, are the values of n_o, e_o, M_DF, #math154#ω_DF, and Ω after deep-space secular and resonance perturbations have been applied.

Here SDP4 calls the periodics section of DEEP which adds the deep-space lunar and solar periodics to the orbital elements (see Section Ten). From this point on, it will be assumed that n, e, I, ω, Ω, and M are the mean motion, eccentricity, inclination, argument of perigee, longitude of ascending node, and mean anomaly after lunar-solar periodics have been added.

Add the long-period periodic terms

#math155#

a_xN = e cosω

#math156#

β = #tex2html_wrap_indisplay4037#

#math157#

I L_L = #tex2html_wrap_indisplay4041#(e cosω)#tex2html_wrap_indisplay4042##tex2html_wrap_indisplay4043##tex2html_wrap_indisplay4044#

#math158#

a_yNL = #tex2html_wrap_indisplay4046#

#math159#

I L_T = I L + I L_L

#math160#

a_yN = e sinω + a_yNL.

Solve Kepler's equation for #math161#(E + ω) by defining

#math162#

U = I L_T - Ω

and using the iteration equation

#math163#

(E + ω)_i+1 = (E + ω)_i + Δ(E + ω)_i

with

#math164#

Δ(E + ω)_i = #tex2html_wrap_indisplay4059#

and

#math165#

(E + ω)₁ = U.

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

#math166#

e cos E = a_xNcos(E + ω) + a_yNsin(E + ω)

#math167#

e sin E = a_xNsin(E + ω) - a_yNcos(E + ω)

#math168#

e_L = (a_xN² + a_yN²)^{#tex2html_wrap_indisplay4068#}

#math169#

p_L = a(1 - e_L²)

#math170#

r = a(1 - e cos E)

#math171#

#tex2html_wrap_indisplay4074# = k_e#tex2html_wrap_indisplay4075#e sin E

#math172#

r#tex2html_wrap_indisplay4077# = k_e#tex2html_wrap_indisplay4078#

#math173#

cos u = #tex2html_wrap_indisplay4080##tex2html_wrap_indisplay4081#cos(E + ω) - a_xN + #tex2html_wrap_indisplay4082##tex2html_wrap_indisplay4083#

#math174#

sin u = #tex2html_wrap_indisplay4085##tex2html_wrap_indisplay4086#sin(E + ω) - a_yN - #tex2html_wrap_indisplay4087##tex2html_wrap_indisplay4088#

#math175#

u = tan^-1#tex2html_wrap_indisplay4090##tex2html_wrap_indisplay4091##tex2html_wrap_indisplay4092#

#math176#

Δr = #tex2html_wrap_indisplay4094#(1 - θ²)cos 2u

#math177#

Δu = - #tex2html_wrap_indisplay4096#(7θ² -1)sin 2u

#math178#

ΔΩ = #tex2html_wrap_indisplay4098#sin 2u

#math179#

Δi = #tex2html_wrap_indisplay4100#sin i_ocos 2u

#math180#

Δ#tex2html_wrap_indisplay4102# = - #tex2html_wrap_indisplay4103#(1 - θ²)sin 2u

#math181#

Δr#tex2html_wrap_indisplay4105# = #tex2html_wrap_indisplay4106##tex2html_wrap_indisplay4107#(1 - θ²)cos 2u - #tex2html_wrap_indisplay4108#(1 - 3θ²)#tex2html_wrap_indisplay4109#

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

#math182#

r_k = r#tex2html_wrap_indisplay4111#1 - #tex2html_wrap_indisplay4112#k₂#tex2html_wrap_indisplay4113#(3θ² - 1)#tex2html_wrap_indisplay4114# + Δr

#math183#

u_k = u + Δu

#math184#

Ω_k = Ω + ΔΩ

#math185#

i_k = I + Δi

#math186#

#tex2html_wrap_indisplay4119# = #tex2html_wrap_indisplay4120# + Δ#tex2html_wrap_indisplay4121#

#math187#

r#tex2html_wrap_indisplay4123# = r#tex2html_wrap_indisplay4124# + Δr#tex2html_wrap_indisplay4125#.

Then unit orientation vectors are calculated by

#math188#

#tex2html_wrap_indisplay4127# = #tex2html_wrap_indisplay4128#sin u_k + #tex2html_wrap_indisplay4129#cos u_k

#math189#

#tex2html_wrap_indisplay4131# = #tex2html_wrap_indisplay4132#cos u_k - #tex2html_wrap_indisplay4133#sin u_k

where

#math190#

#tex2html_wrap_indisplay4135# = #tex2html_wrap_indisplay4136##tex2html_wrap_indisplay4137##tex2html_wrap_indisplay4138#

#math191#

#tex2html_wrap_indisplay4140# = #tex2html_wrap_indisplay4141##tex2html_wrap_indisplay4142##tex2html_wrap_indisplay4143#.

Then position and velocity are given by

#math192#

#tex2html_wrap_indisplay4145# = r_k#tex2html_wrap_indisplay4146#

and

#math193#

#tex2html_wrap_indisplay4148# = #tex2html_wrap_indisplay4149##tex2html_wrap_indisplay4150# + (r#tex2html_wrap_indisplay4151#)_k#tex2html_wrap_indisplay4152#.

A FORTRAN IV computer code listing of the subroutine SDP4 is given below. These equations contain all currently anticipated changes to the SCC operational program. These changes are scheduled for implementation in March, 1981. #center4153#