index_6.html

and escape velocity [plot].

${{v→ -(2^(1/2) G^(1/2) M^(1/2))/R^(1/2)}, {v→ (2^(1/2) G^(1/2) M^(1/2))/R^(1/2)}}$

In[11]:=

$ {{2^(1/2) (G M)/r^(1/2), r>R}, {(G M (-r^2 + 3 R^2))/R^3^(1/2), True}}$

[Graphics:../HTMLFiles/index_89.gif]

Solve[(2 (G M)/r)^(1/2) == vtarget, r]

${{r→ (2 G M)/vtarget^2}}$

Solve[(G M (-r^2 + 3 R^2))/R^3^(1/2) == vtarget, r]

${{r→ -(R (3 G M - R vtarget^2)^(1/2))/(G^(1/2) M^(1/2))}, {r→ (R (3 G M - R vtarget^2)^(1/2))/(G^(1/2) M^(1/2))}}$

In[12]:=

rOFvesc[ve_, R_, G_, M_] := If[ve<11180.7, (2 G M)/ve^2, R (3 G M - R ve^2)/(G M)^(1/2)]

Sqrt[(3 * 6.673 * 10^(-11) * 5.9742 * 10^24)/rE] (* r goes to zero when ve goes to 13693 m/s *)

$Plot[rOFvesc[vesc, rE, 6.673 * 10^(-11), 5.9742 * 10^24], {vesc, 9000, 13693}]$

[Graphics:../HTMLFiles/index_105.gif]

$Table[{i, rOFvesc[i * 1000, rE, 6.673 * 10^(-11), 5.9742 * 10^24]/rE}, {i, 1, 13, 1}] (* for escape velocities of 1 through 13 km/s, earth radii from center are ... *)$

Created by Mathematica (February 13, 2007)