We derive the expression of the residual flux given in Eq. (1), which is valid for a disk seen edge-on, with an intensity vertical profile parametrized by the expression . Such a disk is plotted in Fig. A.1. The vertical scale height h is in this case, constant with the radius, but the following derivation still holds if h varies with the separation to the star, for example if there is a flaring characterized by a flaring index β such that . Similarly, the intensity in the disk mid-plane (z = 0) is constant here and equal to 1, but the derivation still holds if the intensity varies with the separation x.
We consider a point A located at a separation x and a height z. It makes an angle with the disk mid-plane. The vertical profile at this separation as a function of i
displayed in Fig. A.3. The two blue arrows represent the amplitude of field rotation Δθ/2 on each side of A. In cADI, the flux of the reference frame at the position of A is equal to the median of the curve between i − Δθ/2 and i + Δθ/2, sampled according to the parallactic angle distribution. We assumed that this sampling is uniformly distributed. To compute the median, we divide this range into four regions, according to the values of the intensity:
region 1, of length 2i, the farthest on the right characterized by the lowest intensity values;
regions 2 and 3, symmetrical about the mid-plane, of length Δθ/2−2i;
region 4, of length 2i centered on the mid-plane and consisting of the highest intensity values.
This decomposition into four regions of increasing intensity allows us to directly conclude that the median over the four regions is also equal to the median over region 2 only, which is equal to the intensity at point B for Δθ/4.
The disk flux f after subtraction of the reference frame can then be expressed as An experimental verification of this law was performed and is shown in Fig. A.3. The disk flux f in the mid-plane (i = 0) after cADI reduction, expressed in percentage for various disks and field rotations, is plotted against the criterion κc. All curves match well, which confirms the derived theoretical expression. A similar graph can be made for different vertical profiles instead of mid-plane profiles.
Experimental verification of Eq. (1) on simulated disks (with different values of the vertical exponent γ and the vertical height h) for various field rotations Δθ. For κc = 3, 95% of the disk flux f is preserved.
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© ESO, 2012