The Stumpff functions C(z),S(z) developed by Karl Stumpff, are used for analyzing orbits using the universal variable formulation.
clc; clear all; z = -25:0.1:400; n = size(z); for i = 1:n(2) if (z(i) > 0) S(i) = (z(i)^0.5 -sin(z(i)^0.5))/z(i)^1.5; C(i) = (1 - cos(z(i)^0.5))/z(i); elseif (z(i) < 0) S(i) = (sinh((-z(i))^0.5) - (-z(i))^0.5)/((-z(i))^1.5); C(i) = (cosh((-z(i))^0.5) - 1)/(-z(i)); else S(i) = 1/6; C(i) = 0.5; end end % Plot figure(1); scatter(z(1:500),C(1:500),'.b'); hold on;grid on; scatter(z(1:500),S(1:500),'.r'); xlabel('z'); ylabel('C(z),S(z)'); legend('C(z)','S(z)'); title('Stumpff functions C(z),S(z)'); text(10,0.4,'smallsats.org','Color',[0 0 0], 'VerticalAlignment','middle',... 'HorizontalAlignment','left','FontSize',14 ); hold off; figure(2); scatter(z(501:n(2)),C(501:n(2)),'.b'); hold on;grid on; scatter(z(501:n(2)),S(501:n(2)),'.r'); xlabel('z'); ylabel('C(z),S(z)'); legend('C(z)','S(z)'); title('Stumpff functions C(z),S(z)'); text(250,0.0010,'smallsats.org','Color',[0 0 0], 'VerticalAlignment','middle',... 'HorizontalAlignment','left','FontSize',14 ); hold off;
Equations from the book Orbital Mechanics for Engineering Students, Second Edition, Aerospace Engineering
Small Satellites 