In[]:=
Clear[z];​​Clear[M];​​Clear[a];​​Clear[e];​​Clear[r];
In[]:=
f=a*e
Out[]=
ae
In[]:=
B=1/2*ArcCos[-(f^2/r^2+z^2/r^2)+Sqrt[-(4f^2)/r^2+(1+(f^2/r^2+z^2/r^2))^2]]
Out[]=
1
2
ArcCos-
2
a
2
e
2
r
-
2
z
2
r
+
-
4
2
a
2
e
2
r
+
2
1+
2
a
2
e
2
r
+
2
z
2
r

In[]:=
1
2
ArcCos-
2
a
2
e
2
r
-
2
z
2
r
+
-
4
2
a
2
e
2
r
+
2
1+
2
a
2
e
2
r
+
2
z
2
r
;
In[]:=
Theexteriorpotentialoftheoblatespheroidisgivenby:​​
In[]:=
P=3M/(2*f)*(B*(1-r^2/(2*f^2)+z^2/f^2)+r^2/(2*f^2)*Sin[B]*Cos[B]-z^2/f^2*Tan[B])
Out[]=
1
2ae
3M
1
2
1-
2
r
2
2
a
2
e
+
2
z
2
a
2
e
ArcCos-
2
a
2
e
2
r
-
2
z
2
r
+
-
4
2
a
2
e
2
r
+
2
1+
2
a
2
e
2
r
+
2
z
2
r
+
1
2
2
a
2
e
2
r
Cos
1
2
ArcCos-
2
a
2
e
2
r
-
2
z
2
r
+
-
4
2
a
2
e
2
r
+
2
1+
2
a
2
e
2
r
+
2
z
2
r
Sin
1
2
ArcCos-
2
a
2
e
2
r
-
2
z
2
r
+
-
4
2
a
2
e
2
r
+
2
1+
2
a
2
e
2
r
+
2
z
2
r
-
2
z
Tan
1
2
ArcCos-
2
a
2
e
2
r
-
2
z
2
r
+
-
4
2
a
2
e
2
r
+
2
1+
2
a
2
e
2
r
+
2
z
2
r

2
a
2
e
In[]:=
TakethepartialdifferentialofPwithrespecttor:
In[]:=
Fprz=D[P,r];
TakethepartialdifferentialofPwithrespecttoz:
In[]:=
Fpzz=D[P,z];
Foranygivenradiusr,thecorrespondingzcoordinateatthesurfaceoftheoblatespheroidisgivenbytheellipticalrelationshp:
In[]:=
z=Sqrt[(a^2-f^2)*(1-r^2/a^2)]
Out[]=
(
2
a
-
2
a
2
e
)1-
2
r
2
a
In[]:=
andthisexpressionforzissubstitutedintoFprz:​​
In[]:=
Frr=AbsReplaceFprz,
(
2
a
-
2
a
2
e
)1-
2
r
2
a
->z;
andintoFpzz
Fzz=ReplaceFpzz,
(
2
a
-
2
a
2
e
)1-
2
r
2
a
->z;​​​​SimplifiedformofFrr(SeeanswerbyGhoster)
In[]:=
Fr=3/2*(ArcSin[e]-e*Sqrt[1-e^2])/e^3*M*r/(a^3);
SimplifiedformofFzz(SeeanswerbyGhoster)
In[]:=
Fz=-3*(e-Sqrt[1-e^2]*ArcSin[e])/e^3*M*Sqrt[a^2-r^2]/(a^3);
​​"Set parameters for a specific oblate sphere:"​​​​M=2000;a=20;e=0.999
​​"Find the resultant force (Ft) of Fr and Fz:"
In[]:=
Ft=Sqrt[Fr^2+Fz^2];
​​"Find the component (Fc) of the Ft force vector that points at the centre:"
Fc=Abs[FtCos[ArcTan[Fz/Fr]-Pi/2+ArcTan[-r/(Sqrt[(a^2-a^2e^2)*(1-r^2/a^2)])]]];​​
In[]:=
Plot[{Fr,-Fz,Ft,Fc,z},{r,0,19.999}]
Out[]=
​​"Find the diagonal distance R from the centre to surface coordinate:"
In[]:=
R=Sqrt[r^2+(a^2-a^2e^2)*(1-r^2/a^2)];
​​"Find the required tangential velocity u to produce the required centrifugal force assuming F = GMm/R^2"
In[]:=
​​u=Sqrt[M/R];
​​"Find the required tangential velocity (v) to produce the required centrifugal force assuming Fc"
In[]:=
​​v=Sqrt[Fc*R];​​
In[]:=
Plot[{u,v},{r,0,a}]
Out[]=