I'm trying to figure out the error in the opening angle for a cone created with kinematic neutron imaging. The angle is defined as:
$$
\theta = \sin^{-1}\left(\sqrt{\frac{Ep}{E}}\right)\,.
$$
And I want to find the error in this angle. I don't know how to propagate error through an inverse sine function so I made a substitution where:
$$
u = \sin^2(\theta)\,,\qquad u = \frac{Ep}{E}\,.
$$
My work is attached, but my delta-theta at the end doesn't have units of radians or degrees. Where did I go wrong?
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1 Answer
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The rule of thumb is to just take the differential of whatever equation relates the different variables. That tells you how small changes in the variables must be related.
I.e. if your new variable $q$ is related to the old variables $x$, $y$, and $z$ by $q = f(x,y,z)$, then
$$ dq = \frac{\partial f}{\partial x}dx +\frac{\partial f}{\partial y}dy+ \frac{\partial f}{\partial z}dz. $$
I think the key for you will be the differential of $\arcsin(x)$: $$ d \arcsin(x) = \frac{dx}{\sqrt{1-x^2}}. $$ and that of $\sqrt{E_p/E}$: $$ d\sqrt{\frac{E_p}{E}} = \frac{dE_p}{2\sqrt{E_p E}} -\frac{1}{2}\sqrt{\frac{E_p}{E}} \frac{dE}{E} $$
Try using those to expand out the differential of $\theta=\arcsin\left(\sqrt{\frac{E_p}{E}}\right)$.
