# NoisyQUInclination

There is a recent debate about which is the distribution of magnetic field inclinations in the quiet Sun. Some people say that the distribution is close to isotropic and others sustain that the amount of what they call "horizontal" fields is around 5 times larger than the "vertical" fields. I show in this calculation which is the effect of the presence of noise on the inferred value of the inclination if the maximum likelihood solution is used. We recently demonstrated that the maximum likelihood value of the inclination in the weak-field regime is biased towards high values of the inclination and here I show some simple calculations.

Let's first start with a function to return the Stokes parameters in the weak-field regime. In this regime, we find

$$ V(\lambda) = -C\lambda_0^2 g_\mathrm{eff} B_\parallel \frac{\partial I(\lambda)}{\partial \lambda} \\ Q(\lambda) = -\frac{C^2}{4} \lambda_0^4 G_\mathrm{eff} B_\perp^2 \cos 2\phi \frac{\partial^2 I(\lambda)}{\partial \lambda^2} \\ U(\lambda) = -\frac{C^2}{4} \lambda_0^4 G_\mathrm{eff} B_\perp^2 \sin 2\phi \frac{\partial^2 I(\lambda)}{\partial \lambda^2} \\ $$where $C=4.67 \times 10^{-13}$ G$^{-1}$ $\unicode{x212B}^{-1}$

If we define the following merit function

# vsiniDeconvolution

The rotational velocity of stars can be measured with several standard diagnostic methods. However, if the star rotation axis is inclined with respect to the line-of-sight (LOS), the measured velocity is not the equatorial velocity, but

$$ v = v_e \sin i $$where $v$ is the measured velocity, $v_e$ is the equatorial rotation velocity of the star (assuming that the star rotates as a solid body) and $i$ is the inclination angle of the rotation axis with respect to the LOS.

# SpectroPolDeconvolution

# BeyondFillingFactor

It is time to go beyond the concept of filling factors. This is a very simple proposal to this end. Imagine that we gave a certain pixel that we subdivide into $N$ equal-area zones. The polarimetric signal that we measure is given by the addition of the fields of the $N$ zones and the added Gaussian noise with zero mean and variance $$\sigma_n^2$$

$$V = \sum_{i=1}^N \alpha_i V_i + \epsilon.$$