2 Summary of TempMap features and test properties

2.1 Line profile inversion with a penalty function

TEMPMAP recovers the surface temperature distribution from the integral equation that relates the distribution of surface temperature to the observed line profile and light curve variations. Local line profiles are computed from a numerical solution of the equation of transfer with 72 depth points from the grid of model atmospheres published by Kurucz (1993). These model atmospheres are precalculated assuming LTE. The local line profile for each small surface segment is obtained from the grid of profiles by interpolation to match the local effective temperature.

 

 

Table 1: Adopted stellar parameters for the forwardcalculations of profiles

	Case 1	Case 2
$\log g$	4.0	4.0
$T_{\rm phot}$	5000	5000
$v_{\rm e}\sin i$	28	41
Inclination i	65	30
$v_{\rm e}$	31	82
Microturbulence	1.0	1.0
Macroturbulence	3.5	3.5
Abundance	solar	solar

The effects of noise in the data are controlled by a penalty function, or regularizing functional, that prevents the "overinterpretation'' of information contained in the line profiles and the light curve. TEMPMAP incorporates a choice between using a maximum entropy penalty function in solving the inverse problem, or using a Tikhonov penalty function. The form taken by the maximum entropy penalty function is

$$f_{\rm en}= \sum_{i}{{\rm Flux}_{i}{\ln}({\rm Flux}_{i})}.$$ (1)

Here Flux is approximated by ${\omega}T^4$ with ${\omega}$ as an arbitrary constant and T as the local effective temperature normalized by the average effective temperature over the surface of the star. The Tikhonov penalty function is usually represented as

$$f_{\rm t}=\int_{-{\pi}}^{\pi}\int_{-{\pi}/2}^{{\pi}/2}\left... ...rtial \tau}{\partial l}\right)^2\right] {\rm d}{\phi}{\rm d}l$$ (2)

where $\phi$ represents latitude on the stellar surface and l is a linearized measure of longitude. (See Piskunov & Rice 1993 for further description). In practice, the choice between these has little significance because normally only small weighting is given to the penalty function when the noise problem is not serious.

Continuum light variations and their zero point in two bandpasses are also employed by the mapping routine to further constrain the solution.

Figure 1: Artificial input map for the forward data for Case 1 (Case 2 is identical but with different inclination and rotational broadening). The graph shows the input map in Mercator projection (top) and in spherical projection at eight rotation phases (bottom)

Figure 2: Examples of recovery using a Tikhonov penalty function at various S/N ratios. From top to bottom: S/N=3000:1, 300:1, 150:1, and 75:1. The images in the left column are for Case 1, the images in the right column for Case 2. Note that there is no significant difference between the reconstructions once a S/Nratio of approximately 300:1 is surpassed. The reconstruction from S/N=600 (not shown) is practically identical to the S/N=3000 case

Figure 3: The artificial spectral line data (crosses) and their respective fits (lines) for S/N=150:1. Left two panels: Case 1, right: Case 2

Figure 4: The difference maps input-output for S/N=150:1. Left: Case 1, right: Case 2. The grey scale indicates the temperature difference in Kelvin. Note that white regions indicate no difference, and black regions indicate maximum difference. These maps can also be viewed as "sensitivity'' maps for a given input map; the most sensitive areas being the darkest regions

Figure 5: The effects of a constant continuum displacement, $\Delta I$, for all line profiles. Top images with $\Delta I=+0.005$, bottom images with $\Delta I=-0.005$. Shown is only Case 1. The left images were recovered without abundance adjustment while the right images were recovered with abundances adjusted by +0.05 dex (top) and -0.04 dex (bottom). The lower panels show the V and $I_{\rm c}$ light curves (crosses) and their respective fits. From left to right: without abundance adjustment and $\Delta I=+0.005$ and $\Delta I=-0.005$, respectively, and with abundance adjustment and again $\Delta I=+0.005$ and $\Delta I=-0.005$. Note the inability to fit the V-I color if the abundance is not adjusted

2.2 Artificial test data

There are two case situations, i.e. two hypothetical stars resembling typical stellar applications. The stellar parameters for these test stars, called Case 1 and Case 2, are listed in Table 1. They differ only by their rotational velocity and inclination of the rotational axis. We excluded the case of a slow rotator, i.e. $v\sin i<20$ kms , from the tests because that was an issue in a previous paper (Strassmeier & Rice 1998a).

Forward calculations are performed for both cases with the appropriate subroutines of TEMPMAP and used the parameters of the Fe I line at 6411 Å, one of the most widely used spectral line for Doppler imaging. The atomic line parameters were adopted from the Kurucz (1993) line list; with the logarithmic transition probability ($\log gf$) set to -0.35, a lower excitation potential of 3.654 eV, and damping constants that are computed from the classical damping formula from Unsöld (1955) for this forward calculation (more extensively discussed later in Sect. 4.3). Note that the literature provides various values for $\log gf$ of this line. The VALD database (Piskunov et al. 1995; Kupka et al. 1999) lists -0.595, Kurucz (1993) listed -0.820, Lambert et al. (1996) adopted -0.66, and King (1999) obtained -0.717. We have used a larger value than is given in most of the tables. That is because we have found in the past that we needed a larger value even when all the blending lines we thought could be there were taken into account.

The grid spacing on the stellar surface was 5 $\hbox {$^\circ $ }$$\times$5 $\hbox {$^\circ $ }$ , giving a total of 2592 pixels visible if the star is at an inclination of 90 $\hbox {$^\circ $ }$. Figure 1 shows the artificial input map in two commonly used projection styles (Mercator and spherical). The line profiles from the forward calculation are computed for 18 equidistant rotational phases and with 91 equally spaced pixels per profile. The artificial light curve was adopted with 12 equidistant points per bandpass. The artificial light and color curves are not shown in this figure but will appear later in the figures with the test recoveries. Our input map was designed to include the commonly encountered surface structures of active stars like a cool polar spot with an asymmetric appendage, several isolated smaller spots at medium latitude, a two-component circular spot simulating an umbra and penumbra analog at the stellar equator, a hot spot with a temperature of 400 K above the photospheric temperature, and an equatorial band of very weak temperature contrast on one side of the star.

In the following, we compare recoveries with two sources of errors: first, errors in the (artificial) data and, second, errors in the adopted recovery parameters.

Figure 6: As in Fig. 5 but for Case 2 (only the abundance adjusted version is shown). The two images were recovered with abundances adjusted by +0.04 dex (top) and -0.06 dex, respectively