Inversion of the intensity-magnetic field relationship in solar active regions

We discuss the relationship between the EUV spectral line intensities and the photospheric magnetic flux in solar active regions. Since the histograms of the magnetic flux density in active region plages can be approximated by an exponential function, the equation describing how the observed total line intensity integrated over an active region area arises from the magnetic field, can be approximated by a Laplace integral. We use this property to solve an inverse problem and derive a function relating the line intensity from individual loops to the photospheric magnetic flux density at their footpoints. We propose a simple model in which the intensity of a coronal line Fe XVI 360.8 Angstrom in an individual coronal loop is proportional to the footpoint magnetic flux density to the power of delta and explore how well the value of delta is constrained by the observations. Using EUV spectra from the Coronal Diagnostic Spectrometer (CDS) on SOHO and magnetograms from SOHO Michelson Doppler Imager for 26 active regions without sunspots, we find that the value of delta depends on the magnetic flux density threshold used to define active region magnetic area. When even the weakest fields are included, we obtain delta = 1.3, where 1.0 < delta < 1.6 with 90% confidence. This result can be used to provide constraints on coronal heating models.