A comparison of X-ray images of the Sun and full disk magnetograms shows a correlation between the locations of the brightest X-ray emission and the locations of bipolar magnetic active regions. This correspondence has led to the generally accepted idea that magnetic fields play an essential role in heating the solar corona. To quantify the relationship between magnetic fields and coronal heating, the X-ray luminosity of many different active regions is compared with several global (integrated over entire active region) magnetic quantities, The X-ray measurements were made with the SXT Telescope on the Yohkoh spacecraft; magnetic measurements were made with the Haleakala Stokes Polarimeter at the University of Hawaii's Mees Solar Observatory. The combined data set consists of 333 vector magnetograms of active regions taken between 1991 and 1995; X-ray luminosities are derived from time averages of SXT full-frame desaturated (SFD) images of the given active region taken within +/-4 hours of each magnetogram. Global magnetic quantities include the total unsigned magnetic flux Phi(tot) = integral dA\B-z\B-z, tot(2) = integral dAB(z)(2), J(tot) = integral dA\J(z)\, and B-perpendicular to,B- (2)(tot) = integral dAB(perpendicular to)(2), where J(z) is the vertical current density and B-z and B-perpendicular to are the vertical and horizontal magnetic field amplitudes, respectively. The X-ray luminosity L-x is highly correlated with all of the global magnetic variables, but it is best correlated with the total unsigned magnetic flux Phi(tot). The correlation observed between L-X and the other global magnetic variables can be explained entirely by the observed relationship between those variables and Phi(tot). In particular, no evidence is found that coronal heating is affected by the current variable J(tot) once the observed relationship between L-X and Phi(tot) is accounted for. A fit between L-X and Phi(tot) yields the relationship L-X similar or equal to 1.2 x 10(26) ergs s(-1)(Phi(tot)/10(22) Mx)(1.19). The observed X-ray luminosities are compared with the behavior predicted by several different coronal heating theories. The Alfven wave heating model predicts a best relationship between L-X and Phi(tot), similar to what is found, but the observed relationship implies a heating rate greater than the model can accommodate, The "Nanoflare Model" of Parker predicts a best relationship between L-X and B-z,B- (2)(tot) rather than Phi(tot), but the level of heating predicted by the model can still be compared to the observed data. The result is that for a widely used choice of the model parameters, the nanoflare model predicts 1.5 orders of magnitude more heating than is observed. The "Minimum Current Corona" model of Longcope predicts a qualitative variation of L-X with Phi(tot) that agrees with what is observed, but the model makes no quantitative prediction that can be tested with the data. A comparison between L-X and the magnetic energy E-mag in each active region leads to a timescale that is typically 1 month, or about the lifetime of an active region, placing an important observational constraint on coronal heating models. Comparing the behavior of solar active regions with nearby active stars suggests that the relationship observed between L-X and Phi(tot) may be a fundamental one that applies over a much wider range of conditions than is seen on the Sun.
