# Proof and precise formulation of Welch-Satterthwaite equation

In my statistics course notes the Welch-Satterthwaite equation, as used in the derivation of the Welch test, is formulated as follows:

Suppose $S_1^2, \ldots, S_n^2$ are sample variances of $n$ samples, where the $k$-th sample is a sample of size $m_k$ from a population with distribution $N(\mu_k, \sigma_k^2)$. For any real numbers $a_1, \ldots, a_n$, the statistic $$L = \frac{\nu}{\sum_{k=1}^{n} a_k\sigma_k^2}\sum_{k=1}^n a_kS_k^2$$ with $$\nu = \frac{\left(\sum_{k=1}^m a_kS_k^2\right)^2}{\sum_{k=1}^m \frac{(a_kS_k^2)^2}{N_k - 1}}$$ approximately (sic) follows a chi-squared distribution with $\nu$ degrees of freedom.

Doing a quick Google search, most sources seem to follow a similar formulation. There are a few questions I have though:

1. How can the number of degrees of freedom of a chi-squared distribution depend on the statistics $S_i$? Shouldn't the number of degrees of freedom be a constant?
2. What does 'approximately follow a distribution' mean? This is closely related to my third question:
3. How can one justify this approximation? All 'proofs' I have seen on the internet seem to assume that $L$ follows a chi-squared distribution and then show that $\nu$ must have the stated value, using basic properties of expected value and variance. Even those arguments are confusing to mee, since they seem to ignore the fact that the $S_i$ are itself stochastic variables and not known constants.

I found a link to the original papers by Welch and Satterthwaite, which I can't access freely. I wonder if there is a reasonably short answer to at least the first two of my questions. Us students are not expected to be able to prove the equation, so it's not that bad if there is no readily available proof, but I'd like to at least understand the statement itself.