I am trying to find the value of $r_0$ using $L^2=\frac{{r_0}^3 f'({r_0})}{2 f({r_0})-{r_0} f'({r_0})}$ and making a 3d plot with the help of the code given below:
$Assumptions = Thread[{l, L, Q, r0, rp} > 0];
f[r_] := 1 - (2 M)/r + Q^2/r^2 + r^2/l^2;
M = rp/2 (1 + Q^2/rp^2 + rp^2/l^2);
Veff[r_] = f[r] (L^2/r^2 + 1)
eqn = L^2 == (r0^3 f'[r0])/(2 f[r0] - r0 f'[r0]) // Simplify;
sol = Reduce[eqn, r0][[2, 2, 1]] // ToRules;
\[Lambda][Q_, rp_] =
Block[{l = 1, L = 20},
1/2 Sqrt[(r0 f'[r0] - 2 f[r0]) Veff''[r0]] /. sol] // Simplify;
Plot3D[Log[100, (\[Lambda][Q, rp] + 1)], {Q, 0, 1.2}, {rp, 0, 1}]
which gives me root objects as output instead of algebraic expressions.
I tried ToRadicals
as discussed in other related answers but it didn't work for me.
The resulting 3d plot is also blank.
How to solve this problem?
Reference paper (page 8)
Root
objects are roots of 6th order polynomials, where no general solution in terms of radicals exists $\endgroup$$Assumptions = Thread[{l, L, Q, r0, rp} > 0];
the assumptions will automatically be available to any function that uses the optionAssumptions
. After defining$Assumptions
, simplify the equation,eqn = L^2 == (r0^3 f'[r0])/(2 f[r0]-r0 f'[r0])//Simplify;
If you are having trouble plotting, show the code that you used for the plot. $\endgroup$