In 1995 Paulina Lira found, in her Univ. of Chile Master's thesis, that there are uniform B-V colors of unreddened Type Ia supernovae. From 30 to 90 days after the time of V-band maximum the fast decliners and slow decliners alike seemed to give the same locus of B-V vs. time. That would mean that from the observed B-V colors of a Type Ia in the tail of the color curve one could easily determine the B-V color excess of the object. Here is Fig. 1 from Phillips et al. (1999), which is based on photometry with the CTIO 0.9-m telescope.

Of course, having a B-V color excess and knowing the V-band extinction along the line of sight are not exactly the same thing, since there is a scaling factor relating the two, and that scaling factor (known as R_V) is not a universal constant.

Another thing to consider is that to use the Lira Law one has to have photometry converted to the filter system used to establish it. For example, Nick Suntzeff and Mark Phillips found that B-V colors in the tail obtained from imagery taken with the YALO 1-m telescope at Cerro Tololo were roughly 0.1 mag redder compared to B-V colors obtained with 0.9-m photometry. Krisciunas et al. (2003) derived filter corrections for SN 2001el and showed that the systematic differences in B-V color can be attributed to differences in the B and V filters used at the two telescopes. Keep in mind that a supernova does not have the spectrum of a normal star. A month after maximum light it is optically thin at optical wavelengths. While no one would do broad-band photometry of the Orion Nebula, astronomers carry out broad-band photometry of supernovae all the time. So these filter corrections are particularly important once the supernovae become optically thin. In the case of SN 2000cx the published I-band photometry ranges three-quarters of a magnitude a couple months after maximum light!

Consider the graph below, from Krisciunas et al. (2000). Let us consider the hypothetical case of a supernova with V-band extinction A_V = 1.00 mag. With what uncertainty can we derive A_V using photometry from various broad-band filters? The dashed line is the best case scenario using only B- and V-band data. We set A_V = (3.1 +/- 0.1) E(B-V). The dotted line uses A_V = (2.55 +/- 0.3) E(B-V), what Riess, Press, & Kirshner (1996) claim is the most appropriate scaling for galaxies that have hosted Type Ia supernovae. The solid line is the worst case scenario using V-K photometry. We assume the K-band extinction A_K is 0.112 times A_V, but is only known to +/- 50 percent. If there are uniform B-V or V-K colors for Type Ia supernovae, and if the uncertainties of the color excess (i.e. the observed, reddened, value minus the intrinsic color) are greater than 0.02 mag, then one gets a more accurate estimate of the V-band extinction by using V-K colors. Thus, the question is: are there uniform V-K colors or other V minus near-IR colors of Type Ia supernovae?

In the August 20, 2000, issue of the Astrophysical Journal we published a paper in which we asserted that for a range of decline rates of Type Ia SNe, they appear to exhibit uniform color curves (Krisciunas et al. 2000). V-K was a "well behaved" color index, as was V-H. The effect of dust on infrared light is an order of magnitude less than the effect on optical light. As a result, if we could know the V-K color excess, say, we would only have to scale that by 1.129 to obtain the V-band extinction A_V. Even if we dealing with a galaxy with dust very much different than dust in our Galaxy, this factor would probably only range from 1.10 to 1.15. For V-H the scale factor is 1.235 (Cardelli, Clayton, & Mathis 1989).

From our 2000 paper, we show below the V-K curves of two reddened and six unreddened Type Ia SNe. Given the small number of data points per object, we did not feel justified in fitting any more sophisticated a function than two straight lines.

However, from more well sampled light curves such as SNe 1999ac and 2001el we know that there is no abrupt change in the color curve 6 days after the time of B-band maximum. Thus, we now feel justified to fit a higher order polynomial to the data. Below we have subtracted off the V-K reddening of SNe 1999cl and 1998bu and fit a 4th order polynomial to the "compactified" data.

Now, you might be wondering how well this compares with theory. In our paper on SN 2001el, which was submitted to the Astronomical Journal on August 9, 2002, we including some result from modelling by Peter Hoeflich. He finds that because the evolution of the spectrum in the H and K bands is a "well behaved" function of the evolution of the iron lines, and that the V-band is also a nice part of the spectrum, the V-H and V-K color curves should be well behaved. The uncertainty of the modelling is 0.2 to 0.3 magnitudes, and the theoretical color curves can also be shifted +/- 1 day in the time axis. But in this case the agreement of theory and data is actually better than one could expect.

What does this mean? For an object like SN 2001el we derive A_V = 0.57 +/- 0.05 mag from a variety of color indices (B-V, V-I, V-J, V-H, V-K). The uncertainty of the distance to this object and its host galaxy (NGC 1448) is not due to any large uncertainty in the extinction along the line of sight. This is good news. Since one of the more serious sources of systematic error in the determination of astronomical distances is interstellar extinction, we believe we have found a way to diminish the uncertainty in its determination.

One proviso is that our loci shown above are for Type Ia SNe in the mid-range of decline rates. Very slow decliners have bluer loci, and very fast decliners have redder loci. Consider the following graph from Krisciunas et al. (2004). It contains dereddened colors of six Type Ia supernovae with slow decline rates (Delta m_15(B) < 1.01). The unreddened V-H and V-K loci for mid-range decliners are also shown (above the data points in the V-H and V-K diagrams).

In the range of -8 to +9.5 days with respect to T(B_max) the V-J colors get monotonically bluer. The scatter about the straight line is only +/- 0.067 mag. After t = +9.5 days the dispersion increases considerably. Another way of saying this is that slow decliners have uniform V-J colors only in the -8 to +9.5 day window. For V-H the scatter about the fourth order polynomial is +/- 0.062 mag prior to t = +8.5 days. After that the V-H colors are no longer uniform. Our dereddened V-K colors of slow decliners do not exhibit any epoch over which the rms scatter is less than +/- 0.1 mag.

More than one of my colleagues has expressed doubts about the validity of my combining rather irregularly sampled data on two groups of Type Ia supernovae (mid-range decliners and slow decliners) to construct unreddened loci applicable to each group. Type Ia supernovae exhibit a continuous range of temperatures, opacities, and presumably Nickel-56 yields, so shouldn't there be a continuous gradation of V minus near-IR colors as a function of, say, the decline rate parameter Delta m_15(B)? To test this let us pick an epoch that allows us to do interpolation of actual measurements rather than extrapolation of existing data to times when no data were taken. Given the available data, the best epoch to use is +6 days after the time of B-band maximum. Below we show the dereddened V-J and V-H colors of a number of Type Ia SNe at t = +6 days, vs. the decline rate parameter Delta m_15(B).

I think the present data can be interpreted two different ways equally validly. Either we can assert there is a continual variation of V minus near-IR color at t = +6 days, or we can lump the slow decliners (0.8 < Delta m_15(B) < 1.0) together and lump the mid-range decliners (1.0 < Delta m_15(B) < 1.4) together. The only outlier to the latter perspective is the point at Delta m_15(B) = 1.10, V-J = -1.36, which corresponds to SN 1981B.

Krisciunas et al. (2004) give in their Tables 13 and 14 the coefficients necessary to generate the unreddened V minus near-IR loci for slow decliners and mid-range decliners. One can use these loci to determine the V minus near-IR color excesses for Type Ia SNe. Then, using standard reddening coefficients (Cardelli et al. 1989), one can determine the V-band extinction along the line of sight. The equations are:

A_V = (1.393 +/- 0.110) E(V-J) ;

A_V = (1.235 +/- 0.058) E(V-H) ;

A_V = (1.129 +/- 0.029) E(V-K) .

Using a combination of optical and infrared data one can get a much better handle on A_V than can be done solely with optical data. There are two reasons for this: 1) there are sufficiently uniform V minus near-IR colors of Type Ia SNe that one can get meaningful color excesses; and 2) using V-band photometry and one or more near-IR bands, one only has to scale V minus near-IR color excess(es) by a little bit to get A_V. In the case of optical data, A_V = 3.1 E(B-V), but the scale factor 3.1 is only a nominal value for dust in our Galaxy. That scale factor can range from 1.8 to 5. Serious systematic errors of distance can result in the distance determination to a significantly reddened Type Ia SN based solely on B- and V-band photometry. One can largely eliminate large systematic errors by using a combination of optical and IR data.

References:

Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, "The Relationship between Infrared, Optical, and Ultraviolet Extinction," Astrophysical Journal , 345 , 245

Krisciunas, K., Hastings, N. C., Loomis, K., McMillan, R., Rest, A., Riess, A. G., & Stubbs, C. 2000, "Uniformity of (V-Near-Infrared) Color Evolution of Type Ia Supernovae and Implications for Host Galaxy Extinction Determination," Astrophysical Journal , 539 , 658

Krisciunas, K., Suntzeff, N. B., Candia, P., et al. 2003, "Optical and Infrared Photometry of the Type Ia Supernova 2001el," Astronomical Journal , 125 , 166

Krisciunas, K., Suntzeff, N. B., Phillips, M. M., et al. 2004, "Optical and Infrared Photometry of the Nearby Type Ia Supernovae 1999ee, 2000bh, 2000ca and 2001ba," Astronomical Journal , 127 , 1664

Phillips, M. M. 1993, "The Absolute Magnitudes of Type Ia Supernovae," Astrophysical Journal , 413 , L105

Phillips, M. M., Lira, P., Suntzeff, N. B., Schommer, R. A., Hamuy, M., & Maza, J. 1999, "The Reddening-Free Decline Rate Versus Luminosity Relationship for Type Ia Supernovae," Astronomical Journal , 118 , 1766

Riess, A. G., Press, W. H., & Kirshner, R. P. 1996, "Is the Dust Obscuring Supernovae in Distant Galaxies the Same as Dust in the Milky Way?" Astrophysical Journal , 473 , 588

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