index_11.html

Radial distance, velocity and acceleration vs time along the chord

Solve[(R^2 - rf^2)/(rf^2 - rmin^2)^(1/2) == const, rf] (* note to aid solving tgen for rf *)

${{rf→ -(R^2 + const^2 rmin^2)^(1/2)/(1 + const^2)^(1/2)}, {rf→ (R^2 + const^2 rmin^2)^(1/2)/(1 + const^2)^(1/2)}}$

In[24]:=

rf[t_, rmin_, R_, G_, M_] := (R^2 + (Tan[((G M)^(1/2) t)/R^(3/2)])^2rmin^2)/(1 + (Tan[((G M)^(1/2) t)/R^(3/2)])^2)^(1/2)

[Graphics:../HTMLFiles/index_179.gif]

((G M)^(1/2) (-R^2 + rmin^2) Sin[(2 (G M)^(1/2) t)/R^(3/2)])/(2^(1/2) R^(3/2) (R^2 + rmin^2 + (R - rmin) (R + rmin) Cos[(2 (G M)^(1/2) t)/R^(3/2)])^(1/2))

In[25]:=

vf[t_, rmin_, R_, G_, M_] := ((G M)^(1/2) (-R^2 + rmin^2) Sin[(2 (G M)^(1/2) t)/R^(3/2)])/(2^(1/2) R^(3/2) (R^2 + rmin^2 + (R - rmin) (R + rmin) Cos[(2 (G M)^(1/2) t)/R^(3/2)])^(1/2))

[Graphics:../HTMLFiles/index_185.gif]

In[26]:=

(G M (R - rmin) (R + rmin) Sin[((G M)^(1/2) t)/R^(3/2)]^2)/R^3^(1/2)

[Graphics:../HTMLFiles/index_191.gif]

In[27]:=

[Graphics:../HTMLFiles/index_197.gif]

((G M)^(3/2) (R - rmin) (R + rmin) Sin[(2 (G M)^(1/2) t)/R^(3/2)])/(2 R^(9/2) (G M (R - rmin) (R + rmin) Sin[((G M)^(1/2) t)/R^(3/2)]^2)/R^3^(1/2))

In[28]:=

atot[t_, rmin_, R_, G_, M_] := ((G M)^(3/2) (R - rmin) (R + rmin) Sin[(2 (G M)^(1/2) t)/R^(3/2)])/(2 R^(9/2) (G M (R - rmin) (R + rmin) Sin[((G M)^(1/2) t)/R^(3/2)]^2)/R^3^(1/2))

$General :: spell1 : Possible spelling error: new symbol name \"atot\" is similar to existing symbol \"vtot\". More…$

[Graphics:../HTMLFiles/index_204.gif]

Created by Mathematica (February 13, 2007)