# Phillips relationship

For the Phillips relationship in labour-market economics, see Phillips curve.

In astrophysics, the Phillips relationship is the relationship between the peak luminosity of a Type Ia supernova and the speed of luminosity evolution after maximum light. Soviet astronomer Yury P. Pskovskii proposed a peak-luminosity initial decline rate correlation in 1977. American astronomer Bert W. Rust (1974) provided evidence supporting Pskovskii's study. They found that the faster the supernova faded from maximum light, the fainter its peak magnitude was. This result was hard to prove given difficulties in its precise measurement.[1][2][3]

In 1993 Mark M. Phillips discovered a method to prove this relationship precisely during the course of the Calán/Tololo Supernova Survey.[4] The correlation had been difficult to prove because Pskovskii's slope (beta) parameter was difficult to measure with precision in practice, a necessary condition to prove the correlation. Rather than trying to determine the slope, Phillips used a simpler and more robust procedure that consisted in "measuring the total amount in magnitudes that the light curve decays from its peak brightness during some specified period following maximum light." It was defined as the decline in the B-magnitude light curve from maximum light to the magnitude 15 days after B-maximum, a parameter he called $\Delta{m}_{15}$. The relation states that the maximum intrinsic B-band magnitude is given by

$M_\mathrm{max}(B) = -21.726 + 2.698 \Delta m_{15} (B).$

[5]

Phillips dedicated the journal article confirming Yuri Pskovskii's proposed correlation to Pskovskii, who died a few weeks after Phillips' evidence confirming the relationship was published.

It has been recast to include the evolution in multiple photometric bandpasses[6][7] and as a stretch in the time axis relative to a standard template.[8] The relation is typically used to bring any Type Ia supernova peak magnitude to a standard candle value.

The original $\Delta{m}_{15}$ definition drawn by Phillips around 1995.

## References

1. ^ Rust, B. W. "The Use of Supernovae Light Curves for Testing the Expansion Hypothesis and Other Cosmological Relations" (PDF). [PhD thesis, University of Illinois].
2. ^ Pskovskii, Yu. P. (1977). "Light curves, color curves, and expansion velocity of type I supernovae as functions of the rate of brightness decline". Soviet Astronomy 21: 675. Bibcode:1977SvA....21..675P.
3. ^ Pskovskii, Yu. P. (1984). "Photometric classification and basic parameters of type I supernovae". Soviet Astronomy 28: 658–664. Bibcode:1984SvA....28..658P.
4. ^ Phillips, M. M. (1993). "The absolute magnitudes of Type IA supernovae". Astrophysical Journal Letters 413 (2): L105–L108. Bibcode:1993ApJ...413L.105P. doi:10.1086/186970.
5. ^ Rosswog; Bruggen. High Energy Astrophysics.
6. ^ Hamuy, M., Phillips, M. M., Maza, J., Suntzeff, N. B., Schommer, R. A., & Aviles, R. 1995, Astronomical Journal, 109, 1
7. ^ Riess, A. G., Press, W. H., & Kirshner, R. P. 1996, AstrophysicsJournal, 473, 88
8. ^ Perlmutter, S. A., & et al. 1997, NATO ASIC Proc. 486: Thermonuclear Supernovae, 749