We present a method for distance calibration without using standard fitting procedures. Instead we use random resampling to reconstruct the probability density function (PDF) of calibration data points in the fitting plane. The resulting PDF is then
used to estimate distance-related properties. The method is applied to samples of radio surface brightness to diameter (Sigma-D) data for the Galactic supernova remnants (SNRs) and planetary nebulae (PNe), and period-luminosity (PL) data for the Large Magellanic Cloud (LMC) fundamental mode classical Cepheids. We argue that resulting density maps can provide more accurate and more reliable calibrations than those obtained by standard linear fitting procedures. For the selected sample of the Galactic SNRs, the presented PDF method of distance calibration results in a smaller average distance fractional error of up to $approx 16$ percentage points. Similarly, the fractional error is smaller for up to $approx 8$ and $approx 0.5$ percentage points, for the samples of Galactic PNe and LMC Cepheids, respectively. In addition, we provide a PDF-based calibration data for each of the samples.
In this paper we present new empirical radio surface brightness-to-diameter ({Sigma} - D) relations for supernova remnants (SNRs) in our Galaxy. We also present new theoretical derivations of the {Sigma} - D relation based on equipartition or on cons
tant ratio between cosmic rays and magnetic field energy. A new calibration sample of 60 Galactic SNRs with independently determined distances is created. Instead of (standard) vertical regression, used in previous papers, different fitting procedures are applied to the calibration sample in the log {Sigma} - log D plane. Non-standard regressions are used to satisfy the requirement that values of parameters obtained from the fitting of {Sigma} - D and D - {Sigma} relations should be invariant within estimated uncertainties. We impose symmetry between {Sigma} - D and D - {Sigma} due to the existence of large scatter in both D and {Sigma}. Using four fitting methods which treat {Sigma} and D symmetrically, different {Sigma} - D slopes {beta} are obtained for the calibration sample. Monte Carlo simulations verify that the slopes of the empirical {Sigma} - D relation should be determined by using orthogonal regression, because of its good performance for data sets with severe scatter. The slope derived here ({beta} = 4.8) is significantly steeper than those derived in previous studies. This new slope is closer to the updated theoretically predicted surface brightness-diameter slope in the radio range for the Sedov phase. We also analyze the empirical {Sigma} - D relations for SNRs in the dense environment of molecular clouds and for SNRs evolving in lower-density interstellar medium. Applying the new empirical relation to estimate distances of Galactic SNRs results in a dramatically changed distance scale.
The radio surface brightness-to-diameter ({Sigma} - D) relation for supernova remnants (SNRs) in the starburst galaxy M82 is analyzed in a statistically more robust manner than in the previous studies that mainly discussed sample quality and related
selection effects. The statistics of data fits in log {Sigma} - log D plane are analyzed by using vertical (standard) and orthogonal regressions. As the parameter values of D - {Sigma} and {Sigma} - D fits are invariant within the estimated uncertainties for orthogonal regressions, slopes of the empirical {Sigma} - D relations should be determined by using the orthogonal regression fitting procedure. Thus obtained {Sigma} - D relations for samples which are not under severe influence of the selection effects could be used for estimating SNR distances. Using the orthogonal regression fitting procedure {Sigma} - D slope {beta} approx 3.9 is obtained for the sample of 31 SNRs in M82. The results of implemented Monte Carlo simulations show that the sensitivity selection effect does not significantly influence the slope of M82 relation. This relation could be used for estimation of distances to SNRs that evolve in denser interstellar environment, with number denisty up to 1000 particles per cm3 .
An analysis of the relation between radio surface brightness and diameter, so-called Sigma-D relation, for planetary nebulae (PNe) is presented: i) the theoretical Sigma-D relation for the evolution of bremsstrahlung surface brightness is derived; ii
) contrary to the results obtained earlier for the Galactic supernova remnant (SNR) samples, our results show that the updated sample of Galactic PNe does not severely suffer from volume selection effect - Malmquist bias (same as for the extragalactic SNR samples) and; iii) we conclude that the empirical Sigma-D relation for PNe derived in this paper is not useful for valid determination of distances for all observed PNe with unknown distances.
