Extinction and spectral slope

In a recent work we dereddened the observed VRI magnitudes of V404 Cygni assuming Av=4.04, Rv=3.1, and the CCM extinction law. Spectral slopes between V- and I-bands were calculated using:

(1)
\begin{align} \alpha = \frac{log(f_V/f_I)}{log(\nu_V/\nu_I)} \end{align}

where $f_{V}$ and $f_{I}$ are flux-densities in V- and I-bands respectively. Also, here $\nu_{V}$ and $\nu_{I}$ are the central frequencies of the V- and I-bands, which are 550.3 THz and I: 375.7 THz respectively.

According to CCM, $A_I = 0.48 A_V$. Therefore $A_V=4.04$ implies $A_I = 1.94$.

If $A_V$ were 10% less, then $A_I$ would also be 10% less, and the star would be intrinsically brighter by 0.4 mag in V-band and 0.19 mag in I-band. In flux-density units this would mean that the star is brighter by a factor of $c_V = 10^{0.4/2.5}$ in V-band and a factor of $c_I = 10^{0.19/2.5}$ in I-band. The new spectral slope $\alpha_{new}$ in this case can be written as

(2)
\begin{split} \alpha_{new} &= \frac{log\frac{f_{V;new}}{f_{I;new}}}{log(\nu_V/\nu_I)}\\ & \\ &= \frac{log\frac{c_V * f_{V;old}}{c_I * f_{I;old}}}{log(\nu_V/\nu_I)}\\ & \\ &= \frac{log(c_V) - log(c_I) + log\frac{f_{V;old}}{f_{I;old}}}{log(\nu_V/\nu_I)}\\ & \\ &= \frac{log(c_V) - log(c_I)}{log(\nu_V/\nu_I)} + \alpha_{old}\\ & \\ &= \frac{0.4 - 0.19}{2.5log(\nu_V/\nu_I)} + \alpha_{old}\\ & \\ &= 0.507 + + \alpha_{old}\\ \end{split}

Therefore the spectral slope would be 0.5 higher if the reddening was 10% less. Following the same set of equations we see that if the reddening was instead 10% higher then the slope would be 0.5 lower.

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