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

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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.

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We face the problem of correcting two-dimensional spectropolarimetric data from the perturbation introduced by the PSF of the Hinode solar optical telescope. The two-dimensional data has been obtained by scanning a slit on the surface of the Sun and recording the information of the four Stokes profiles $(I,Q,U,V)$ on each point along the slit for a set of discrete wavelength points around the 630 nm Fe \textsc{i} doublet. As a consequence, the data can be considered to be four three-dimensional cubes of images. We use the notation $\mathbf{I}(\lambda)$, $\mathbf{Q}(\lambda)$, $\mathbf{U}(\lambda)$ and $\mathbf{V}(\lambda)$ to refer to observed images at a certain wavelength $\lambda$. In practice, given the scanning process, these are not strictly speaking images, because each column of the image is taken at a different time. In general, in the standard image formation paradigm, the observed image $\mathbf{I}$ (for simplicity we focus on Stokes $I$ but the same expressions apply to any Stokes parameter) that one obtains in the detector after degradation by the atmosphere and the optical devices of the telescope at a given wavelength can be written as: $$ \mathbf{I} = \mathbf{O} * \mathbf{P} + \mathbf{N}, \label{eq:image_formation} $$

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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 $$\sigma_n^2$$

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

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