I have a 5 order polynomial equation which gives log of a chi factor required to convert equivalent widths to normalised H alpha luminosity for M dwarfs, (Reiners et al. 2008). I would like to get the error on this chi factor but my error propagation method is giving an error ~ 7 times the chi factor.
The formula is as such; $log(\chi) = a + bT_{eff} + cT_{eff}^2 + dT_{eff}^3 + eT_{eff}^4 + fT_{eff}^5$
which by taking an exponential on both sides gives; $\chi = e^a \times e^{bT} \times e^{2cT} \times e^{3dT} \times e^{4eT} \times e^{5fT}$ (T is $T_{eff}$ here)
Using $f = ae^{bA}$ ---> $\sigma_{f} \approx |f||b\sigma_{A}|$ I get;
$\sigma\chi = |\chi||\sigma_{T_{eff}}|(b + 2c + 3d + 4e + 5f)$
Can someone please help me in where I'm going wrong.
You need to keep the powers of $T$ in your exponents; $e^{cT^2} \ne e^{2cT}$, and similarly for the other powers. It may be simpler to stay with the first form, and find the uncertainty on $\log \chi$.