# [SOLVED]data from 2nd order ode mathematica

#### dwsmith

##### Well-known member
How can I extract time data from a system 2nd order ODEs in Mathematica?

#### Ackbach

##### Indicium Physicus
Staff member
Can you post the ODE's and your Mathematica code so far?

#### dwsmith

##### Well-known member
Can you post the ODE's and your Mathematica code so far?
Code:
Numerical Simulation of the 2-Body Problem

ClearAll["Global*"];

We begin with of the the necessary data for this problem....

M = 5974*10^21; (* mass of Earth, kg *)

m = 1000; (* mass of spacecraft, kg *)

\[Mu] = 3.986*10^5; (* gravitaional parameter, based on km units of length, \
km/s for velocity *)

Rearth = 6378; (* radius of the Earth, km *)

Simulation Inputs

r0 = {3950.55, 43197.9, 0};(* initial position vector, km *)
v0 = {3.3809, -7.25046, 0}; (* initial velocity vector, km *)
Days = 1/10; (* elapsed time of simulation days *)

\[CapitalDelta]t = Days*24*3600;(* convert elapsed days to seconds *)

s = NDSolve[
{
x1''[t] == -(\[Mu]/(Sqrt[x1[t]^2 + x2[t]^2 + x3[t]^2])^3)*x1[t],
x2''[t] ==  -(\[Mu]/(Sqrt[x1[t]^2 + x2[t]^2 + x3[t]^2])^3)*x2[t],
x3''[t] ==  -(\[Mu]/(Sqrt[x1[t]^2 + x2[t]^2 + x3[t]^2])^3)*x3[t],

x1 == r0[], (* intial x-position of satellite *)

x2 == r0[],(* intial y-position of satellite *)

x3 == r0[],(* intial y-position of satellite *)

x1' == v0[],(* intial vx-rel of satellite *)

x2' == v0[],(* intial vy-rel of satellite *)

x3' == v0[](* intial vy-rel of satellite *)

},
{x1, x2, x3},
{t, 0, \[CapitalDelta]t}
];

Plot of the Trajectory Relative to Earth

g1 = ParametricPlot3D[
Evaluate[{x1[t], x2[t], x3[t]} /. s], {t, 0, \[CapitalDelta]t},
PlotStyle -> {Red, Thick}];
g2 = Graphics3D[{Blue, Opacity[0.6], Sphere[{0, 0, 0}, Rearth]}];

Show[g2, g1, Boxed -> False]`

#### BillSimpson

##### New member
I'm not certain exactly what you are expecting when you "extract time data", but perhaps something in this will help.

What if you replace

Show[g2, g1, Boxed -> False]

with

g3 = Graphics3D[Table[Point[{x1[t], x2[t], x3[t]} /. s[]], {t, 0, Δt, Δt/10}]];
g4 = Graphics3D[Table[Text[ToString[t], {x1[t], x2[t], x3[t]} /. s[], {1, 0}], {t, 0, Δt, Δt/10}]];
Show[g2, g1, g3, g4, Boxed -> False]

That will place points along your path and label each point with the associated value of t. You can adjust the step size and the label position next to each point as needed.

If that doesn't provide you with the necessary level of detail then you might try using

Table[{t,x1[t],x2[t],x3[t]}/.s[], {t,0,Δt,Δt/10}]//TableForm

with appropriate step size to give you the numerical results to go along with your plot.