Robust Constraints

Here we discuss the constraints imposed by the radio observations for SN2003L. As a preliminary constraint on the total energy of the source, we estimate the brightness temperature (Tb) of the supernova and compare it with the robust constraints imposed by equipartition arguments and the inverse Compton catastrophe (ICC). From [6], the brightness temperature is defined as a function of the observed flux density, the peak frequency, and the angular size of the source. As an initial estimate for the physical size of the SN ejecta, we first assume the optical expansion velocity of 5900 km s_1 [8] can be used as an average speed to describe the motion of the radio bright ejecta. Using an approximate explosion date of 2003 Jan 1 UT based on optical light-curves [7], we estimate the shock radius to be r « 4.3 x 1015 cm at t « 85 days when the observed flux density peaked at 8.5 GHz. Using the observed peak flux of F8.5 GHz ~ 2.8 mJy and adopting a source distance of 91.7 Mpc (Qm = 0.27, QA = 0.73, H0 = 71 km s^1 Mpc^1), we find a brightness temperature of Tb « 1.8 x 1012 K which is dangerously near the ICC limit of Tb < 1012 K. This suggests that the radio ejecta expanded with a velocity significantly higher than that observed at optical frequencies. In fact, an ejecta velocity of ~ 16,000 km s_1 3 times larger than that derived from optical spectroscopy) would be necessary to avoid violating the ICC constraint.

Assuming equipartition of energy places a further constraint on the brightness temperature limit and reduces it to Tb < 5 x 1010 K. Using the equipar-tition arguments of [6] and [3], we derive the minimum energy for the radio supernova. Assuming that the observed radio flux is produced by synchrotron emission, the total energy of the source (U) can be expressed as the sum of the energy in relativistic electrons (Ue) and the energy in the magnetic field (Ub). At equipartition, the fraction of total energy in electrons equals the fraction of total energy in magnetic fields (ee = eB = 1) and the total energy is minimized at Ueq [6]. This occurs when the emitting source reaches an equipartition radius denoted by the angular size, 9eq. The minimum energy of the source can be thus be parameterized in terms of the synchrotron peak flux and the equipartition size.

Using our most densely sampled light-curve, we fit for the peak flux (Sp) over the observed ~ 200 day evolution and find Sp « 2.8 mJy at vp = 8.5 GHz on 2003 Mar 27 UT (« 85 days since explosion). For this epoch we estimate an angular size 0eq « 19^as (with ¡3 « —1.0) which implies an average shock velocity of v « 0.1c and an equipartition brightness temperature of Tbeq « 5.0 x 1010 K. By setting U = Ueq we find the energy is minimized at the equipartition value of Ueq « 4.3 x 1047 erg with an associated magnetic field strength of BPeq « 0.6 G. As shown by [6], synchrotron emission systems which diverge from equipartition necessitate a huge increase in total energy. Consequently, it is possible that the total energy contained within the fast moving ejecta of SN2003L is in fact much larger than 4.3 x 1047 ergs.

These preliminary constraints allow us to make two robust conclusions:

1. The velocity of the radio bright SN ejecta must be at least 16, 000 km s_1 to avoid violating the inverse Compton catastrophe limit and ~ 30,000 km s_1 assuming equipartition.

2. The energy of the supernova must be > 4.3 x 1047 erg and could be significantly larger depending on the proximity of the system to equipartition.

These conclusions imply that there was a considerable amount of energy released at high velocities in the type Ic supernova explosion of SN2003L. In fact, these equipartition constraints alone demand that SN2003L is among the

120 Alicia M. Soderberg 105

Fig. 2. The diversity of peak luminosities and observed time of the peak is shown above for all radio bright type Ib/c supernova. By assuming equipartition, the expansion velocity for each event can be estimated (dashed lines). Note the radio ejecta of SN2003L is relatively slow and luminous.

most energetic type Ib/c supernovae observed to date, second only to the unusual event of SN1998bw/GRB980425. Fig. 2 is a compilation of all the radio bright type Ib/c supernovae observed to date. By comparing the peak radio luminosity to the observed time of peak flux, the diversity in equipartition derived expansion velocities can be examined. Note that although SN2003L peaks later, it is among the brightest radio supernovae.

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