1 Introduction

Radiometers on board satellites launched during the last three decades (NIMBUS-7, SMM, UARS, EURECA, SOHO) have revealed that the total solar irradiance, also referred to as the solar constant, changes on a variety of time-scales. Solar irradiance variations on scales of days up to the solar activity cycle length are closely related to the evolution of the solar surface magnetic field, because the emergence and evolution of active regions (AR) on the solar surface is reflected in the irradiance records (Lean et al. 1998; Fligge & Solanki 2000b). Sunspots and active region faculae are considered to be the dominant contributors to solar irradiance changes on time-scales of days to weeks. Space-based irradiance records have also established a variation of about 0.1% of the irradiance in phase with the 11 year solar activity cycle, giving as a result a brighter Sun around activity maximum (Chapman 1987; Willson & Hudson 1988). The origin of the long-term increase of the irradiance between activity minimum and maximum is still widely debated. It is expected that small-scale magnetic elements that compose the enhanced and quiet network contribute substantially to the observed irradiance increase during activity maximum (Foukal & Lean 1988; Solanki & Fligge 2001; Fligge & Solanki 2000a). Nevertheless, other mechanisms of non-magnetic origin have also been proposed, based, for example, on temporal changes in the latitude-dependent surface temperature of the Sun (Kuhn et al. 1988). Other authors have tried to explain these variations by modelling structural changes in the convection zone during the solar cycle (Balmforth et al. 1996).

The photospheric magnetic field is concentrated in discrete elements whose diameters range from less than a hundred to several tens of thousands of kilometers. The brightness signature of these magnetic features is a strong function of their heliocentric angle and their size; sunspots are dark while small flux tubes are generally bright; faculae appear brighter near the limb (Solanki 1993, 2001). However, our knowledge of the brightness of small scale magnetic features, groups of which form faculae and the network, is incomplete (e.g. Solanki 1994).

Faculae, bright structures seen in the photosphere cospatially with chromospheric plages, are associated with magnetic fields. At high resolution, they consist of many unresolved small continuum bright points, with diameters of about 100 km, called facular points (Muller 1983; Berger et al. 1995). The observed zoo of small magnetic features is unifyingly described by the concept of the small flux tube. To predict their physical properties, different models for small flux tubes have been constructed (e.g., Spruit 1976; Deinzer et al. 1984a, 1984b; Knölker et al. 1988; Knölker & Schüssler 1988; Grossmann-Doerth et al. 1989; Steiner et al. 1996). According to this model faculae are conglomerates of evacuated flux tubes with hot walls and a hot or cool floor (corresponding to an optical depth of $\tau=1$) depending on the evacuation and diameter of the flux tube. The model predicts a certain CLV of the contrast for a particular diameter of the underlying flux tubes.

This model assumes that inside each small flux tube the magnetic field is of the order of a kilogauss, but practically zero outside. Due to the magnetic pressure the flux-tube interior is evacuated, so that $\tau=1$ is reached along its walls (which are bright due to radiation leaking in from the surroundings, i.e., there is a horizontal flux of energy into the tube). Flux tubes can be dark at disk center if suppression of convective energy transport within the tube is included (e.g., Spruit 1976; Deinzer et al. 1984a, 1984b; Knölker et al. 1988), resulting in a cooling of the deeper layers. When the tube is sufficiently broad, the horizontal optical depth between the wall and the tube center is large and most of the radiation cannot reach the center. In this case, the tube floor remains dark at its center. But if the tube is sufficiently slender, the horizontal flux of energy can reach the center of the tube; then, the interior of the tube is heated, the vertical energy flux increases, and the tube turns bright, even at disk center. In this scenario, the transition between smaller bright points and larger dark micropores (e.g., Topka et al. 1997) occurs at a diameter of about 300 km (e.g., Grossmann-Doerth et al. 1994), and therefore micropores would fill the gap between small bright points and larger dark pores. Micropores are predominantly found in active regions, while bright points are the main constituents of the network. When observed near the limb, the heated walls of the tube become visible, and therefore the contrast increases.

Observations also provide evidence that the contrast, as well as the underlying thermal structure, depends on the size of the flux tubes (e.g., Keller 1992), but more commonly on the strength of the magnetogram signal (Frazier 1971; Spruit & Zwaan 1981; Solanki & Stenflo 1984; Solanki 1986; Zayer et al. 1990; Solanki & Brigljevic 1992; Topka et al. 1992, 1997; Lawrence et al. 1993; Grossmann-Doerth et al. 1994), and hence a test of flux tube models is possible with such data. However, since most flux tubes are not resolved, it is necessary to have a well-defined and constant spatial resolution of the observations with which to compare the models. With ground-based data, the basis of practically all facular contrast CLV measurements to date, this criterion is hard to meet.

Differences in spatial resolution, caused by variable seeing, may indeed partly explain the variety of measured contrast CLVs (e.g., Libbrecht & Kuhn 1984; Unruh et al. 1999). Other possible factors are differences in wavelength, spectral resolution and the magnetic filling factor of the observed features (Solanki 1994). The problem posed by variable seeing can be circumvented by employing data recorded in space, while the magnetic filling factor can be estimated with the help of cospatial and cotemporal magnetograms. Only relatively few contrast investigations including the magnetogram signal can be found in the literature (e.g. Frazier 1971; Foukal & Fowler 1984; Topka et al. 1992, 1997; Lawrence et al. 1993).

Here we add to this list using data from the MDI instrument on board SOHO (Domingo et al. 1995); their main advantages are:

• seeing effects due to the Earth's atmosphere are avoided,
• measurements are made in the pure continuum (i.e. spectral resolution is not a problem),
• the 20-min averaged MDI magnetograms have a reasonably low noise level,
• a large and homogeneous data set is available,
• the characteristics of the instrument and the data sets are well known and stable,
• magnetograms and intensity images are obtained regularly by the same instrument with exactly the same spatial resolution, so that a one-to-one identification of brightness with magnetogram signals can be made.

The main disadvantages of the MDI data are:

• a relatively low spatial resolution with a pixel size of $2\times2\arcsec$,
• measurements are available at only a single wavelength.

The purpose of this paper is to present new high-quality measurements of the contrast of the photospheric bright features as a function of both heliocentric angle and magnetogram signal and to obtain an analytical function that predicts their contrast given a position on the disk and a magnetic signal value. Such measurements are expected to be of use not only to constrain models of flux tubes, but also to improve the modelling of the solar irradiance (Lean et al. 1998). Uncertainties in the contrast of faculae and the network are one of the major sources of error in the modelling of solar irradiance variations. Employing MDI data to obtain the contrast as input for irradiance modelling is of particular interest since MDI magnetograms have already been succesfully used for such modelling (Fligge & Solanki 2000a).

In Sect. 2 we present the data sets used and the analysis procedures. In Sect. 3 we describe the results, which are discussed in Sect. 4. Finally, our conclusions and a summary are given in Sect. 5.