To derive the number of SNe expected in our ground-based data and calculate the IR supernova rate, we have to derive a relation between the luminosity of the galaxies and their Star Formation Rate (SFR) and the shape of the "mean" K-band light curve for a core-collapse SN.

The K-band light curves of SNe

To constrain the infrared SN rate we have to compare the number of detections with the number of expected events. The latter number can be derived from the observation log and the limiting magnitude once the amount of time for which the SNe remain above the  detection limit (control time) is known for each galaxy. Therefore it is critical to have a good knowledge of the NIR light curve.
We have collected 22 K-band light curves of SNe with a well known epoch of maximum light, for a total of more than 200 IR observations (see Mannucci et al. 2003 for details).

The fraction of the slow declining events appears to be not negligible: 6 out of 22 SNe show a peculiar behaviour at some point of their evolution, and 5 of them are classified as type IIn SNe from the optical spectra. The fraction is even higher, 6 out of 22, if only the SN with observations after 4 months from the maximum are considered. Such an high fraction is certainly due to selection effects, as SNe with NIR excess are preferentially observed at such late times. This brightening or stabilization of the luminosity at NIR wavelength is usually attributed to thermal emission from dust forming in the ejecta or present in the pre-existing circumstellar medium. The heating could be due to the interaction between the ejecta and the circumstellar medium. For this reason this effect is expected to be more common in the events occurring in the dense, star forming regions,
as supported by the observation of the bright radio emission from the SN remnants in the starburst galaxies. These SNe becomes much brighter than the others, but their constant luminosity make their detection by variability methods almost impossible if the moment of the explosion is not observed.
The remaining  "ordinary events'' show inhomogeneous behaviour during the first 4 months. Most of them show a maximum 10-20 days after the optical peak, and a linear decline afterward. The NIR luminosity peak of core-collapse SNe is similar to that of the type Ia, computed to be -18.8 at day=25 (Meikle, 2000), even if the core-collapse SNe are usually much fainter in the optical. SN  1999em is the less luminous event, about 1 mag fainter at the optical maximum than any other measured SN, and shows a very delayed peak, occurring at about epoch=70 days.  The spread between the SNe of each class is comparable with the spread between the classes, i/e., there is no evidence of systemic differences for the core-collapse SNe of different sub-types. This is probably due to the fact that the light curve depends not only on the properties of the progenitor but also on the circumstellar medium. After the first 120 days, all the ``ordinary'' SNe show a similar  declining rate of about 0.025 mag per day.  This decline rate is much faster than the decline rate of about 0.010 observed at these late time in the optical in the type Ib and II SNe and expected from the ⁵⁶Co decay (0.0098 mag/day). It is also faster than the optical decline rate of the type Ia SNe (about 0.015 mag/day).

The spread of the absolute magnitude near the peak is about 3 magnitude, and about 1.5 magnitude at later times. To take into account for this large spread we have defined the upper and lower envelops (in gray) of the distributions and will use these curves to compute the maximum and minimum SN rate expected from the SFR. The upper envelop is defined by SNe 1979C and 1982R, the lower envelop by SN 1999em.
As average light-curve we used SN 1980K, an event showing a linear decline and a peak magnitude near the average value. We preferred to use an observed event instead of an average between the two envelops, strongly dependent on peculiar sub- or super-luminous events and not corresponding to any observed SN.

The L(FIR) - SN rate relationhe K-band light curves of SNe

Several authors have computed the SN rates as a function of the host galaxy morphological type. These estimates are based on optical observations of local galaxies having dust contents and SFRs much lower than the objects in our sample. Rates are generally normalized to the luminosity of the parent galaxy, either the B luminosity of L(FIR). The former normalization is implied by the common use of the SN unit (SNu), i.e., the number of SNe per century per 10¹⁰ L(SUN) of B luminosity. This luminosity has an ambiguous physical meaning, as in active galaxies is roughly proportional to the SFR, in the quiescent galaxies is related to the total mass, and in the dusty starburst galaxies in very sensitive to the dust content and distribution. For galaxies with a significant SFR it is more useful, for many applications and especially when the core-collapse SNe are considered, to normalize the observed number of SNe to the FIR luminosity, known to be proportional to the SFR. Several indirect methods based on radio, FIR or mid-infrared observations have been used derive the SN rate in obscured starburst galaxies, as described in van Buren \& Greenhouse (1994) and Mattila & Meikle (2000).  These authors have derived the relation between the SNr and L(FIR) using galaxies of low FIR luminosity, and therefore a large extrapolation is needed apply these result to our sample. We have extended the observed range toward high luminosity by using the relation between radio luminosity and SNr in Condon & Yin (1990) and data  in Hackenberg et al. (2000)  and Wilson et al. (1999). In this way we obtain the relation:

The IR Supernova Rate

Using this relation and the mean K-band light curve to compute the control time of the observations, we derive an expected number of SNe from our ground-based sample of 18 SNe, assuming that the 80% of the FIR luminosity of the galaxies is related with the nuclear regions where our sensitivity is reduced by the subtraction residuals. This number has to be compared with the 4 detected SNe.
The first conclusion is that NIR searches for SN in starburst are, as expected, more efficient than similar searches but at optical wavelengths, as at least part of the SN are heavily dusty. In fact, from the B luminosity of the galaxies we were expecting only 0.5 Sne, and two of the detected SNe were in fact only observed in the NIR. Maiolino et al. (2002) demonstrated that at least SN 2001db was too absorbed to be detected in the optical even at its maximum. This higher SN rate reflects the higher extinction affecting the B light (which is the normalizing factor of SNu) of the galaxies in our sample, their enhanced star formation and the higher efficiency of the NIR observations with respect to the optical.

The second conclusion is that the major fraction of SNe expected from the FIR luminosity are still missing, i.e., we have detected only about 25% of the expected SNe.
This smaller number of detections can be explained in several ways:

•  The most probable scenario is that most of the SNe are so embedded into the dust that their luminosity is vastly reduced even at near-IR wavelengths. We have computed the average extinction needed to reduce the expected off-nuclear number to the observed number, founding A_V>25 mag with a formal confidence level of 90%. These values for the extinction are similar to those obtained by other authors. In this case, NIR search for SN can detect only a fraction of the total SN rate, and a radio survey should also be planned.

• If 100% of the FIR flux comes from the central arcsec, only 12 events are expected because of the residuals present in the nuclear region even after applying a PSF matching algorithm. When considering also the uncertainties in the predictions, dominated by the dispersion in luminosity of the SNe, the number of expected SNe is compatible with the observed one without the need of a large additional extinction. The presence of a large number of SNe in the central arcsec of the starburst galaxies can be tested by near-IR monitoring from space: NICMOS on HST can provide images with higher resolution (0.2") and the more stable PSF to considerably reduce the central residuals and reveal the nuclear SNe. We have obtained time on NICMOS for such a monitoring.

Less likely possibilities are the presence of obscured AGNs dominating the FIR flux of the galaxies, an environment dependence of the SN luminosity and taht the relation between L(FIR) and SN rate could be overestimating the SN rate for a given FIR luminosity.

To conclude, the infrared SN rate that we obtain from our data is given by: