Threaded Mode | Linear Mode

Eddie W. Shore · 08-13-2017, 01:13 PM

Link to the blog post: http://edspi31415.blogspot.com/2017/08/h...nates.html

The following program tracks the position (x,y) of a satellite in orbit by solving the pair of second-order differential equations:

d^2 x/dt^2 = -α * x * (x^2 + y^2)^-1.5

d^2 y/dt^2 = -α * y * (x^2 + y^2)^-1.5

For planetary orbits, α = G * M

Variables and Labels

Labels Used: 4 (A, B, C, D)

Variables Used:
Pre stored variables:
X = x0, initial x position
U = dx/dt, velocity of x coordinate
Y = y0, initial y position
V = dy/dt, velocity of y coordinate
C = Δt, desired time increment
A = α, gravitational force factor

If you are analyzing planetary orbits, then α = G * M where M is the mass of the center object and G is the Universal gravitational constant, G = 6.67384 * 10^-11 (m^3)/(s^2 * kg).

Other:
T = t

This program is based off of the HP 25S and program authored Robert M. Eisberg. (see the source below)

HP 32SII Program (87.5 bytes)

Code:

A01 LBL A   // start of program 

A02 0

A03 STO T   // initialize time variable

C01 LBL C

C02 RCL Y

C03 RCL Y

C04 x^2

C05 RCL X

C06 x^2

C07 +

C08 -1.5

C09 y^x

C10 RCL A

C11 *

C12 *       // calculate –α*y*(x^2 + y^2)^-1.5

C13 LASTx  

C14 RCL X

C15 *      // calculate –α*x*(x^2 + y^2)^-1.5

C16 RCL T

C17 x=0?

C18 GTO B

C19 R↓ 

D01 LBL D

D02 RCL C

D03 *

D04 STO- U

D05 R↓ 

D06 RCL C

D07 *

D08 STO- V

D09 RCL C

D10 STO+ T

D11 RCL U

D12 *

D13 STO+ X

D14 RCL X

D15 STOP       // view X

D16 RCL C

D17 RCL V

D18 *

D19 STO+ Y

D20 RCL Y

D21 STOP       // view Y

D22 GTO C      // start a new loop

B01 LBL B

B02 R↓ 

B03 2

B04 ÷ 

B05 x<>y

B06 2

B07 ÷ 

B08 x<>y

B09 GTO D

During execution, recall T for the time variable. Run by XEQ A.

Example

Pre stored variables:
X = 1.5
U = 0.18
Y = 1.5
V = 0.23
C = 1
A = α = 1

Table:
T = X = Y =
0 (initial) 1.5 1.5
1 1.6014 1.6514
2 1.5713 1.6672
3 1.4105 1.5443
4 1.0956 1.2527
5 0.5429 0.6892
6 -0.8138 -0.8949

Source: Robert M. Eiseberg. Applied Mathematical Physics with Programmable Calculators McGraw Hill. 1976. ISBN 0-07-019109-3