The Physics of Supernova 1987A
Richard McCray
Abstract We describe multiwavelength observations of the evolving spectra and images of Supernova (SN) 1987A, and we review the principles used to infer the physical conditions in the explosion debris. We interpret the early optical and gamma- ray light curves with a simple diffusion model. We review the evidence for dust formation in the debris. We show X-ray and optical observations that enable us to characterize and map the shock fronts caused by of the interaction of the debris with circumstellar matter. We describe how observations of millimeter emission lines due to rotational transitions of CO and SiO enable us to map the distribution, masses, and temperatures of these molecules in the debris.
Contents 1 Introduction...... 2 2 Supernova Energetics...... 3 3 The Light Curve...... 4 4 X-Rays and Gamma Rays...... 6 5 Spectral Evolution...... 7 6 Dust Formation...... 8 7 Circumstellar Matter...... 9 8 The Impact...... 10 8.1 Plane-Parallel Shocks...... 10 8.2 Blast Wave and Reverse Shocks...... 12 9 Radiation from Shocked Gas...... 14 9.1 X-Ray Emission from Non-radiative Shocks...... 14 9.2 Radiative Shocks: The Hotspots...... 19 9.3 Balmer-Dominated Shocks: The Reverse Shock...... 20 9.4 Doppler Tomography...... 21 10 Interior Debris...... 23 10.1 Molecular Emission from Inner Debris...... 23 10.2 Rotational Transitions of CO and SiO...... 25
R. McCray () Department of Astronomy, University of California, Berkeley, CA, USA e-mail: [email protected]
© Springer International Publishing AG 2017 1 A.W. Alsabti, P. Murdin (eds.), Handbook of Supernovae, DOI 10.1007/978-3-319-20794-0_96-1 2 R. McCray
10.3 Modeling the Spectral Line Energy Distribution...... 26 Cross-References...... 29 References...... 29
1Introduction
Supernova 1987A in the Large Magellanic Cloud was first observed on February 23, 1987. It is the brightest supernova since Kepler’s supernova of 1604. By virtue of its proximity, SN is the first supernova to be observed at every band of the electromagnetic spectrum and the first to be observed through its initial flash of neutrinos. SN1987A was classified as a Type II supernova, i.e., its spectrum was dominated by hydrogen lines. But it had an unusual light curve, which continued to brighten for about three months after its initial outburst before it began to fade. We now understand that this behavior was a consequence of the fact that its progenitor was a blue giant rather than a red giant star. Modern supernova surveys show that roughly 1Ð2% of Type II supernovae display light curves and spectral evolution similar to SN1987A. SN1987A is surrounded by a system of three circumstellar rings (Fig. 1), which evidently were ejected by the progenitor some 20,000 years before the supernova outburst. The cylindrical symmetry of the ring system strongly indicates that the progenitor of SN1987A was a binary star system. The absence of any evidence for a surviving companion star suggests that the two stars merged before the explosion,
Fig. 1 SN1987A as seen with the Hubble Space Telescope in 2010 (Courtesy of Peter Challis) The Physics of Supernova 1987A 3 and this scenario may account for the facts that the progenitor was a blue giant and that it ejected the triple ring system. Today, almost 30 years after its discovery, SN1987A has made the transition to the supernova remnant phase, in which its luminosity is dominated by emission from shocks formed where the supernova blast wave encounters the inner circumstellar ring. Although the supernova debris has faded by a factor 10 7, it is still observable at wavelengths ranging from radio to gamma rays, and it continues to be the most intensively observed supernova in history. In this chapter, I aim to provide a simple framework for understanding the physics of this evolution. To do this, I will rely on rough order-of-magnitude estimates. The reader can find many references to the observations and more details of their interpretation in review articles by Arnett et al. (1989), McCray (1993) and McCray and Fransson (2016).
2 Supernova Energetics
SN1987A is a core-collapse supernova, which means that the explosion is the result of the gravitational collapse of the iron core of a massive star. In the final stages of evolution, thermonuclear reactions in the core convert H to He, He to C, O, Ne, Si, and ultimately to Fe. Lacking any source of thermonuclear energy, the iron core cools and shrinks until it is supported by degeneracy pressure of electrons. At this point the core resembles a white dwarf star, having mass comparable to Mˇ, the mass of the Sun, and radius RC 1000 km. The mass of the core continues to increase as a result of thermonuclear reactions in shells surrounding the core. When the core mass exceeds MC D 1:4 Mˇ (the Chandrasekhar limit), electron degeneracy pressure can no longer withstand the pull of gravity. The core collapses 3=2 1=2 on the free-fall timescale, tff RC .GMC / 2 ms. The collapse is halted at a 15 3 radius Rcore 12 km, at which point the density of the core, core 10 gcm , is comparable to that of an atomic nucleus and the nuclear force becomes strongly 2 53 repulsive. The kinetic energy of infall, EG .3=5/GMC =Rcore 3 10 ergs, is converted to heat, which is divided equally among photons, electrons, positrons, and three species of neutrinos and antineutrinos. The temperature of the resulting fireball 4 3 can be estimated from the relationship EG D .43=8/aT Œ4 Rcore=3 , which yields 3=4 kT 100 MeVŒRcore=10 km . At such a temperature and density, the iron nuclei dissolve into neutrons and protons. Most of the protons are converted to neutrons by inverse beta decay. The result is a nascent neutron star. During the first few seconds after the collapse is halted, the hot neutron star is opaque, even to neutrinos. The neutrinos carry the internal heat by convection to the neutrinosphere, the surface above which the neutrinos can freely stream outward. The convection is violently unstable, and this instability reverses the infall of matter into the neutron star and deposits ESN 1%ofEG as thermal and kinetic energy of outflowing matter. This outflowing matter acts as a piston, driving a shock wave through the envelope of the star. Thermonuclear reactions in the shocked gas synthesize heavy elements, including the radioisotopes 56Ni, 57Ni, and 44Ti. 4 R. McCray
Hydrodynamic instabilities during the first few days after the explosion cause the heavy elements to be mixed with the O, C, Ne, Si, etc., that were synthesized during the late stages of evolution of the progenitor star and with the hydrogen/helium of the stellar envelope. This mixing is macroscopic, not microscopic, in the sense that the fragments of different chemical composition do not interact chemically except at their boundaries. Most of the binding energy of the newly formed neutron star emerges as a burst of neutrinos lasting several seconds. In the case of SN1987A, this neutrino flash was detected through flashes of Cerenkov light seen in deep underground tanks of water in Japan and Ohio. The total energy of the neutrino flash was E 5 52 10 ergs, as expected for the formation of a neutron star. (E D EG =6, because the gravitational collapse energy was divided equally among six types of neutrinos, only one of which was detected). The duration, 10 s, and the characteristic temperature of the neutrinos, kT 4 MeV, were also just as predicted for the formation of a neutron star. As mentioned, a small fraction of the energy of the neutrinos is deposited as thermal and kinetic energy in a hot bubble of gas that reverses the flow of matter falling toward the nascent neutron star. This bubble, which resides within the neutrinosphere at 100 km, drives a shock wave through the envelope of the star. The passage of the shock deposits half the energy of the bubble as kinetic energy of the exploding star and half as thermal energy. The thermal energy resides mostly as photons. One can estimate the temperature of the radiation by equating 3 4 51 7 3 1=4 .4 R =3/aT0 D 1:5 10 ergs, which gives T0 10 K ŒE51=R12 , where 51 12 E51 D ESN =10 ergs and R12 is the radius of the progenitor in units 10 cm. We may estimate the characteristic expansion velocity, V , of the supernova 2 debris by equating the explosion energy, ESN D MV =2, which yields V D 1=2 1 3200ŒE51=M10 km s , where M10 is the debris mass in units of 10 solar masses. The time for the supernova blast to propagate through the debris is given by 1=2 1=2 t0 D R=V 1 h R12M10 E51 .
3 The Light Curve
Figure 2 shows light curves of various components of SN1987A. The optical display of the supernova will commence when the blast arrives at the photosphere, an event called shock breakout. According to the (very rough) estimate above, this event should occur about 1 h after the neutrino flash. At shock 6 breakout, the temperature of the photosphere will suddenly rise to TS 10 K and subsequently decrease rapidly (within a few hours) as the debris expands. The shock breakout yields a flash of ionizing radiation having initial luminosity 45 L0 10 ergs=s and total fluence of ionizing (>13:6 eV) radiation Fi D 2 1057 photons. The net energy of ionizing radiation released at shock breakout, 47 Ei 10 ergs, is negligible compared to the total energy of the supernova explosion, 51 ESN 3 10 ergs. The Physics of Supernova 1987A 5
Fig. 2 SN1987A Light Curves (McCray and Fransson 2016). Solid curves are debris: green Ð radioactive deposition; violet Ð far infrared; cyan Ð optical. Dashed curves are equatorial ring: pink Ð radio (3Ð20 cm); green ÐX-rays (0.5Ð3 keV); red Ð UV/optical; gold Ð near infrared (5Ð30 m)
For the first few months, the supernova envelope is opaque to the radiation deposited in the debris by the blast wave, so that most of the radiation produced by the supernova blast cannot escape freely. The opacity is dominated by Compton scattering, and one may estimate roughly the optical depth by assuming that the 3 supernova envelope has uniform electron density, ne D M=ŒmH 4 R =3 .The 24 2 optical depth is given by D ne T R, where T D 0:67 10 cm is the 9 2 Thomson scattering cross section. Accordingly, we estimate 2 10 M10R12 . The characteristic time for radiation to escape from such an envelope is given by 11 1 tesc R =c 0:7 10 s M10R12 . The radiation will remain trapped until a time 7 3=4 1=4 t D tesc,ort 1:5 10 s M10 E51 , or about 4.6 months for M10 D 1 and E51 D 3. This estimate agrees fairly well with the observed time, t D 3 months, of maximum light for SN1987A. By maximum light, the characteristic radius of the debris will have increased to 3 1=4 1=4 R12 4:3 10 M10 E51 (about 380 AU for E51 D 3). The trapped radiation cools according to the adiabatic law, Prad / , where the adiabatic index is D 4=3 for 3 4 4 a radiation-dominated fluid. Since / R , Prad / R . Given that Prad / T , 1 1 we find that T / R and the energy of trapped radiation decrease as Erad / R . By the time that the trapped radiation can escape, its energy will have diminished by a factor 4:3 103. As the fireball expands, the radiation pressure causes the expanding debrisp to accelerate, so that the characteristic expansion velocity increases by a factor 2. If the explosion were the only source of energy in the supernova debris, the optical display would be very faint, with total energy <1048 ergs, far less than 6 R. McCray the observed value. But the debris contains another source of energy: radioactivity of newly synthesized isotopes, principally 56Ni and its daughter 56Co. The fast positrons and gamma rays from the decay of these isotopes deposit their energy in the debris in timescales comparable to their mean lifetimes, tNi D 8:76 dfor 56 56 Ni and t C o D 111:3 dfor Co. Assuming that the gamma rays are absorbed, the radioactive energy deposited by 56Ni is 1.75 MeV per decay and, by 56Co, 3.83 MeV per decay. The total energy deposited in the supernova envelope by 56Ni 50 56 50 is ENi D 0:6 10 MNi ergs and that by Co is ECo D 1:3 10 MNi ergs, where 56 MNi is the initial mass (in Solar units) of Ni produced by the supernova. The energy deposition rate by these isotopes is, therefore, D 0:8 1044 ergs/s 43 MNi exp. t=tNi/ C 1:36 10 ergs/s MNiŒ1 exp. t=tNi/ exp. t=tCo/ .By 7 months after the explosion, tesc t and the bolometric luminosity of the supernova will be equal to the nuclear energy deposition, now dominated by 56Co decay. A fit to the observed bolometric luminosity for 7 months 4 X-Rays and Gamma Rays The decay of 56Co produces gamma ray emission lines at 847 and 1238 keV, and these were observed as early as 190 days Sunyaev et al. (1987). As McCray (1993) describes, these observations imply that a significant amount of 56Co resides at Thomson optical depths as low as T 1, whereas the Thomson optical depth from the center of the debris has the value 25 at 190 days according to our simple uniform density model with E51 D 3 and M10 D 1. The fact that some of the newly synthesized radioactive elements lie in the outer portion of the debris is consistent with three-dimensional hydrodynamic simulations (Fig. 3)ofthe explosion that show plumes of 56Ni extending beyond the inner debris, where most of the nucleosynthesis products reside. For t>2years, the supernova envelope became transparent to gamma rays. Thereafter, most of the gamma rays emerged without depositing their energy in the supernova debris. In that case, the energy deposition by radioactive elements is dominated by stopping of the fast positrons that accompany their decay. (It is The Physics of Supernova 1987A 7 56 Fig. 3 Isocontours of the Ni distribution for a 20 Mˇ (left) and 15 Mˇ (right) progenitor 16 h after explosion (Wongwathanarat et al. 2015). The color bars give the expansion velocity of the surface. Note the higher velocities in the strongly fragmented distribution for the 15 Mˇ model. The heating from the 56Ni decay has not been included generally assumed that the positrons do not propagate far from the place where they are emitted because a very weak magnetic field is sufficient to trap them.) By t D 4:25 years, the abundance of 56Co had decreased by a factor 10 6, and the radioactive energy deposition was dominated by less abundant isotopes with 57 longer mean lifetimes, notably Co (t57 D 271 days). Today, the radioactive energy 44 deposition is dominated by the decay (electron capture) of Ti (t44 D 86:7 years), which is accompanied by the emission of gamma ray lines at 67.9 and 78.3 keV. The 44 C 44 Ti decay is followed by the prompt .tSc44 D 5:7 h) ˇ decay of its daughter Sc, which deposits 0.73 MeV of positron kinetic energy. Observations of the gamma ray lines by the INTEGRAL (Grebenev et al. 2012) and NUSTAR (Boggs et al. 2015) 4 44 observatories enable us to infer the mass, M44 D .1:5 ˙ 0:5/ 10 Mˇ,of Ti and the net energy deposition due to its decay. The green curve in Fig. 2 shows the energy deposition in the supernova debris due to 56Ni, 56Co, 57Co, and 44Ti. 5 Spectral Evolution For the first few months after the explosion, the supernova debris was opaque. Accordingly, its optical spectrum was dominated by a continuum. Blackbody fits to the continuum give a temperature Tph 5500 K, which remained fairly constant in time. During this photospheric phase of spectral evolution, the radius of the photosphere can be inferred from the observed luminosity according to the equation 2 4 Lph D 4 Rph Tph. A fit to the evolution of Lph shows that Rph increased as 8 R. McCray 1=2 Rph / t to a maximum value Rmax 100 AU by tmax 3 months. Note that the velocity, Vph D dRph=dt, decreases with time, indicating that the photosphere was moving inward with respect to the debris. At tmax, the radial velocity of the matter at the photosphere was Vmatter D Rmax=tmax 1800 km/s. After this time, the photosphere shrank rapidly as the optical continuum faded. This event marked the transition from the photospheric phase to the nebular phase, in which the radiation from the supernova debris was dominated by emission lines. Even during the photospheric phase, the optical continuum of SN1987A was punctuated by emission lines, notably H˛,[OI] 6300; 6364 [Ca II] 7291, 7324, and Ca II 8600.TheH˛ line had a P-Cygni profile, consisting of bright redshifted emission and a blueshifted absorption feature, evidently caused by resonant scat- tering of continuum photons by excited hydrogen atoms in the rapidly expanding debris above the photosphere. As the supernova continued to expand, the emission lines became narrower, indicating that the fast-moving gas was becoming less dense and fading, while the photosphere moved inward to reveal emission from more slowly expanding matter deeper within the supernova debris. The widths of the emission lines during the nebular phase implied that most of the line emission was confined to a spherical volume expanding with radial velocity 1800 km/s. McCray (1993) gives a detailed discussion of the evolution of the nebular spectrum. Here, we only summarize the most important conclusions from the analysis of this spectrum. First, the supernova debris cooled rapidly, owing to adiabatic expansion and radiative cooling. For example, infrared emission bands from vibrationally excited CO molecules indicated that the gas emitting these bands cooled from T 4000 K at 192 days to T 1800 K at 377 days. Recent (Kamenetzky et al. 2013) observations of rotational transitions of CO made with the ALMA observatory show that the emitting gas has continued to cool to T < 100 K. Second, different temperature evolution histories were inferred from emission lines of different elements, clear evidence that the debris was not mixed at the microscopic level but instead remained fragmented into chemically distinct regions. 6 Dust Formation A dramatic change in the light curve and spectrum of SN1987A occurred in the interval 250 days t 550 days. First, the optical light curve began to fade more rapidly than the 111.3 day exponential decay. Second, a far-infrared (FIR) ( >8 m) continuum appeared. Third, at the same time, the red sides of the optical and near-infrared (NIR) emission lines from the interior debris began to vanish. All these observations could be explained by the formation of dust grains in the inner debris. The dust was evidently black, in the sense that there was no preferential extinction of the optical lines compared to the NIR lines. Moreover, the dust continuum could be fit better with a Planck spectrum rather than the emission spectrum of typical interstellar dust grains. The fact that the dust blocked almost the The Physics of Supernova 1987A 9 entire red sides of the emission line profiles from the interior debris but roughly half of the blue sides indicated that the dust was confined to opaque clouds that blocked almost all the light from the far hemisphere of the debris and half of the light from the near hemisphere. By 600 days, the FIR continuum from dust was equal to 30%ofthe bolometric luminosity of the supernova debris. The temperature of the dust was T 600 K at 400 days and decreased to T 140 K at 1316 days. At these early times, the minimum mass of dust required to account for the optical and NIR 4 extinction was 3 10 Mˇ. Twenty-eight years after the explosion, an observation with the Herschel obser- vatory (Matsuura et al. 2011, 2015) discovered a strong FIR continuum, which dominated the bolometric luminosity of SN1987A. A model fit to the continuum spectrum gave a dust temperature Td 17 23 K and a minimum mass of 0:4 Mˇ of silicate dust (Dwek and Arendt 2015) An image of the continuum source taken with the ALMA (Indebetouw et al. 2014) confirmed that most of the dust is confined to the inner debris. The fact that the extinction by dust obscures most of the optical and NIR emission from the inner debris makes it difficult to infer anything about nucleosynthesis yields or the morphology of the inner debris from optical or NIR spectra taken after the dust began to form. The only way to see the entire debris is to observe it at sub-mm wavelengths, at which the dust becomes transparent. 7 Circumstellar Matter At t 80 days, narrow (FWHM 30 km s 1) emission lines of N V 1238, 1242, N IV] 1486, N III] 1750, C III] 1909, He II 1640, [O III] 4363, [O III] 5007, and a few other ions appeared in the spectrum of SN1987A. These emission lines brightened to a maximum at t 400 days and then began to fade. Fransson et al. (1989) recognized that these lines came from nearly stationary gas surrounding the supernova that was photoionized by the initial flash of EUV radiation from the supernova. From the time to reach maximum, they inferred that the circumstellar matter was located at a distance 0:6 lt-years from the supernova. From the rate of fading of the emission lines they could infer that most of the 4 3 emitting gas had electron density ne .2 4/ 10 cm , but the persistence of [O III] 5007 required an additional component of circumstellar gas with lower 3 3 density, ne 10 cm . The structure of the circumstellar matter became clear in 1990, when images obtained with the ESO NTT and the HST showed that the line emission was coming 18 from a circular ring of diameter dr D .1:27˙0:07/ 10 cm that was inclined with i D 43ı. Assuming that the ring was thin and uniform, Dwek and Felten (1992) constructed a model for the light curves of the narrow emission lines. The model consists of a geometrical model for the illumination of the ring by the supernova flash convolved with physical models for the decay of emission lines from flash- ionized gas in the ring (Lundqvist and Fransson 1991, 1996; Mattila et al. 2010; see also Sect. 9). This model fits the data qualitatively but not quantitatively (Gould 10 R. McCray 1995). By comparing the angular diameter of the ring with its physical diameter as inferred from the light curves of the narrow lines, one can infer that the distance to the supernova is D D 46:8 ˙ 0:8 kpc. This distance estimate relies on an assumption that we now have reason to doubt. That is, that the narrow emission lines come from the same gas as is observed in the image seen by the HST. We now know, from observations of the impact of the blast wave with the ring, that the ring is not uniform but consists of some thirty hotspots of high-density gas protruding inward from a substrate of lower-density gas. When the ring is illuminated by the ionizing flash, most of the narrow line emission comes from the hotspots. But, since the timescale for fading of the line emission is inversely proportional to the gas density, the hotspots vanish first. By the time that the HST images became available, the image was dominated by the lower- density gas, which has angular diameter about 5% greater than the hotspots. To be consistent, one should compare the image of the hotspots with the light curves from the narrow emission lines. One can estimate that correction will yield a distance about 5% greater than the estimate above. The cylindrical geometry of the triple ring system suggests that rotation plays a key role in its formation, but the actual mechanism remains uncertain. Proposed scenarios range from mass loss from a rapidly rotating single star (Chita 2008)to the merger of a binary star system (Morris and Podsiadlowski 2007). An important 1 clue comes from the radial velocity of the ring, vr D 10:3 km s . Assuming that the equatorial ring has been expanding at constant velocity, we can estimate the time when it was expelled, t D dr =2vr 20;000 years. 8 The Impact As discussed above, most of the energy of the supernova explosion resides as kinetic energy of the expanding debris. This energy becomes manifest when the debris strikes circumstellar matter. The collision creates a system of shocks that suddenly slow down the debris while accelerating the circumstellar matter. The shocks suddenly heat the gas, causing it to radiate. The radiating system resulting from this impact comprises the supernova remnant. The circumstellar matter surrounding SN1987A has a complex structure (Fig. 4). Consequently, the system of shocks resulting from the impact of the debris of SN1987A with this matter is complex, and the radiation from such shocks has a complex spectrum. To interpret the observations, we review the basic physics of plane-parallel shocks, following Ryden (http://www.astronomy.ohio-state.edu/~ dhw/A825/notes7.pdf). 8.1 Plane-Parallel Shocks In a frame of reference in which the shock is stationary, the gas entering the shock from upstream is characterized by its density, 1, velocity, u1, pressure, P1 D nkT1, and internal energy density, 1. The corresponding parameters The Physics of Supernova 1987A 11 Fig. 4 Schematic diagram of SN1987A and its inner circumstellar ring, viewed normal to the equatorial plane. The nucleosynthesis products in the interior debris are confined mostly within a comoving sphere expanding with velocity 1800 km s 1.Theblue represents the outer envelope of the supernova, which is composed mostly of hydrogen and helium. The blue-yellow interface represents the reverse shock, while the yellow annulus represents the X-ray-emitting gas, bounded on the outside by the blast wave. The white fingers represent protrusions of relatively dense gas. As the blast wave overtakes these fingers, they light up as hotspots describing the downstream flow are 2, u2, P2, and 2. The internal energy density of the gas is related to its density and pressure by D P=Œ. 1/ . The adiabatic index is D 5=3 for a monatomic nonrelativistic gas and D 4=3 for a relativistic gas (such as a radiation field). The Rankine-Hugoniot equations representing conservation of mass, momentum, and energy across a plane-parallel shock are: 1u1 D 2u2; (1) 2 2 1u1 C P1 D 2u2 C P2; (2) and 2 2 u1=2 C 1 C P1= 1 D u2=2 C 2 C P2= 2 (3) We define the shock Mach number as the ratio of the velocity of the upstream flow divided by its sound speed: 12 R. McCray 2 1=2 1u1 M1 D (4) P1 In terms of the Mach number, the Rankine-Hugoniot equations yield C 1 2= 1 D u1=u2 D 2 (5) . 1/ C 2=M1 and 2 2 1u1 1 P2 D 1 2 (6) C 1 M1 A strong shock is the limit M 1, and hence P1 is negligible. In that case we find (for D 5=3) 2 D 4 1; (7) u2 D u1=4 (8) and 3 kT D u2 (9) 2 16 1 where is the mean molecular weight per particle. 8.2 Blast Wave and Reverse Shocks The simplest model for the propagation of the blast from a supernova explosion into uniform interstellar gas is the famous Sedov solution. In this solution, the radius, R, of the blast wave depends on only three parameters: E, the energy of the explosion; , the density of interstellar gas; and t, the time since the explosion. The only combination of these three parameters that has the dimensions of length is R D AŒE= 1=5t 2=5, where A is a dimensionless parameter. A detailed solution of the hydrodynamic equations gives A D 1:17. The Sedov solution is based on the assumption that the energy of the explosion resides entirely in the interstellar gas overtaken by the blast wave. It is a good approximation for the propagation of a thermonuclear explosion in the Earth’s atmosphere but not for young supernova remnants. In the latter, a substantial fraction of the hydrodynamic energy resides in the expanding debris of the explosion. In the simplest case, in which both the supernova debris and the circumstellar matter are spherically symmetric and have power-law radial distributions, the impact results in two shocks, as illustrated in Fig. 5. The shock that propagates forward into The Physics of Supernova 1987A 13 Fig. 5 Simulation of the hydrodynamics of a supernova striking circumstellar gas. Gray scale represents gas density. The expanding outer atmosphere has a power-law density distribution / v 9t 3. Left panel: in the polar direction the structure obeys the similarity solution described by Chevalier (1982). In the equatorial direction the blast wave is decelerated suddenly when it encounters a toroidal distribution of relatively dense gas and a reflected shock merges with the reverse shock. Right panel: the blast wave has encountered the dense equatorial ring, sending a transmitted shock into the ring and another reflected shock backward, where it merges with the reverse shock the circumstellar matter is called the blast wave, while the shock that propagates back into the debris is called the reverse shock. Sandwiched between these two shocks is a double layer consisting of shocked supernova debris on the inside and shocked circumstellar gas on the outside. The boundary between these two layers is called the contact discontinuity. We may characterize the density distribution of the circumstellar medium as s c / r , where s D 0 for uniform density and s D 2 (for a steady wind.) Models for the hydrodynamics of a supernova explosion show that the outer debris will have 3 n a power-law density distribution, d .r; t/ / t .r=t/ , where the index n ranges from n D 7 (Chevalier 1982)ton D 9:6 (Ensman and Burrows 1992). In this case, it is possible to construct a self-similar solution for the propagation of shocks (Chevalier 1982). Self-similarity implies that the ratio of the radius of the blast wave to that of the reverse shock is constant and that the density of the circumstellar matter at the blast wave is a constant fraction of the density of the supernova debris at the reverse shock. Assuming so, we find from dimensional analysis that the radius of .n 3/=.nCs/ 2=3 the blast wave evolves as RB / t (e.g., RB / t for n D 9 and s D 0). Chevalier presents detailed solutions of the structure of the similarity solutions for n ranging from n D 6 to n D 14 and both s D 0 (uniform density circumstellar matter) and s D 2 (steady stellar wind). For these solutions the ratio of the radius of the blast wave to that of the reverse shock is typically 1:2. Moreover, the density of the shocked debris at the contact discontinuity is greater than that of the shocked circumstellar gas, so that the contact discontinuity will be unstable in the decelerating flow. 14 R. McCray If, as is the case with SN1987A, the supernova debris strikes a more complex distribution of circumstellar gas, a more complex distribution of shocks will ensue. In particular, if the blast wave strikes a density discontinuity in the circumstellar gas, it will split into two shocks: a transmitted shock that propagates into the denser gas and a reflected shock that propagates backwards into the gas behind the blast wave. If the density contrast is great, the dense gas acts as a rigid obstacle. In that limit, one may show from Eqs. (1), (2), (3), (4), (5), (6), (7), (8), and (9) that the reflected shock propagates backward with twice the velocity of the blast wave, compressing the gas by a further factor 2:5 and raising the temperature by a factor 2:4. 9 Radiation from Shocked Gas We may define three categories of shocks in supernova remnants: non-radiative shocks, radiative shocks, and Balmer-dominated shocks. 9.1 X-Ray Emission from Non-radiative Shocks In non-radiative shocks, the timescale for energy loss by radiation exceeds the characteristic timescale for the system, so that the hydrodynamics is unaffected by the radiation. Actually, the term non-radiative shock is a misnomer. Radiation at X-ray wavelengths is commonly observed from non-radiative shocks in supernova remnants. According to Eq. (9), a shock entering stationary gas with velocity u1 D 1 1000 km s V1000 will heat the gas to a temperature 2 kT D .1:95 keV/V1000; (10) where we have set D 1 for a composition of 50% hydrogen and 50% helium by number, appropriate for the circumstellar matter around SN1987A. Accordingly, gas behind shocks moving with velocity 300 < V < 3000 km s 1 will be heated to temperature 0:176 keV < kT < 17:6 keV. Such gas will radiate primarily in the soft X-ray band. The radiation from such a gas will be dominated by electron impact excitation of emission lines. The luminosity of each emission line may be calculated from Li;j;k.T / D EMAi fj Ci;j;kEi;j;k; (11) where the volume integral Z EM D dVnen (12) The Physics of Supernova 1987A 15 is called the emission measure. To calculate the complete spectrum, one should add a similar expression for continuum emission due to bremsstrahlung, recombination, and excitation of metastable levels that decay by 2-photon emission. The Ei;j;ks are the energies of photons emitted following excitation of ions ni;j D ni fi;j having fraction fi;j in ionization state j , and Ai is the fractional abundance of element i, ni D Ai n. The functions Ci;j;k.T / are the rate coefficients for electron impact excitation of states Ei;j;k, i.e., Z 3=2 2 2 Ci;j;k D .2 kT =me/ 4 v dvexpŒ mev =kT v .v/ (13) where the i;j;ks are the electron impact excitation cross sections. References to values of i;j;k and Ci;j;k for hundreds of ions can be found in Sutherland and Dopita (1993). To complete the task of calculating the emission spectrum of hot gas, it’s necessary to calculate the distribution of ionization states for each element. This is accomplished by solving coupled rate equations for ionization and recombination of each element: d X X f D n f I .T / C n f I .T /; (14) dt j e j j;k e k k;j k k where the quantities Ij;k.T / are rate coefficients for the reactions .nj Ce ! nk Ce, which include impact ionization to states k>jplus radiative and dielectronic recombination to states k X L.T / D Li;j;k D nen .T /; (15) i;j;k where the second equality shows explicitly the fact that L.T / is proportional to the square of the gas density. The function .T /, called the cooling function, is independent of density and is illustrated in Fig. 7. 16 R. McCray Fig. 6 Population of ionization states of neon in coronal ionization equilibrium (Landini and Monsignori Fossi 1990) Fig. 7 Radiative cooling function (Sutherland and Dopita 1993). The curves are labeled by log10 [Fe]/[H], where [Fe]/[H]D 1 corresponds to solar system abundance ratio nFe=nH D 3 10 5.Thedashed curve labeled CEI represents the coronal ionization equilibrium model (with [Fe]/[H]D 1), while the other curves represent NEI models The Physics of Supernova 1987A 17 To account for departures from CIE in modeling the X-ray emission from shocked gas, one can solve the coupled differential equations (Eq. 10), assuming that the gas entering the shock has relatively low ionization. Therefore, accurate models for the emission of radiation from shocked gas require the simultaneous solution of the coupled differential equations (Eq. 10). Assuming that the gas enters the shock at relatively low ionization and that its temperature remains constant downstream from the shock, the solutions will depend on two parameters, temperature, T, and ionization age, net. One may then calculate the spectrum of the shocked gas by integrating the line emissivity (Eq. 11) along the downstream flow from the time that the gas first entered the shock to the present. This procedure yields a set of non-ionization equilibrium or NIE, models for fi .T; nt/ and Li;j;k.T; nt/. Although the CIE approximation provides much physical insight into the radi- ation from a hot gas, it can be off the mark when calculating the radiation from shocked gas. The error arises from the fact that electron impact ionization rates and radiative recombination rates are typically slow compared to excitation rates. As a result, the ionization balance tends to lag changes in temperature. Gas suddenly heated to a given temperature on passage through a fast shock will be less ionized at that temperature than dictated by the CIE approximation. Typically, lower ionization states have more bound levels that can be excited by thermal electron collisions. Consequently, a suddenly heated gas will be under-ionized compared to the CIE model, and its actual line emission can be substantially greater than that given by the CIE model. Conversely, if a gas at a given temperature is allowed to relax by radiative cooling from an initial state of CIE, it will be over-ionized, so that the emission of radiation will be less than given by the CIE model. Therefore, accurate models for the emission of radiation from shocked gas require the simultaneous solution of the differential equations (Eq. 14) coupled with the appropriate fluid equations for the evolution of density and temperature. For 105 K We can derive an important clue to the shock hydrodynamics from the profiles of the X-ray emission lines, which have FWHM 500 km/s. According to Eq. (9), to heat gas to temperature 2 keV requires a shock having velocity 1000 km/s. But the gas is not moving this fast. This apparent paradox is resolved when we recognize that the hottest gas has been shocked a second time by a reflected shock from a dense obstacle. The reflected shock simultaneously slows the gas down as it elevates the temperature and density. The low temperature peak in the bimodal distribution function probably comes from shocks that have been transmitted into clumps of intermediate-density gas as they are overtaken by the blast wave. As Fig. 7 shows, the cooler gas will cool more rapidly than the hotter gas. In this case, the radiative cooling may be rapid enough to modify the hydrodynamics of the shocked gas, as described below. 9.2 Radiative Shocks: The Hotspots In radiative shocks, energy loss due to radiation modifies the hydrodynamics of the shocked gas. Unlike non-radiative shocks, radiative shocks convert a substantial fraction of the thermal energy of the shocked gas into optical radiation. The optical hotspots in SN1987A are due to radiative shocks. After passage of the shock, the hot gas begins to cool and compress as a result of emission of radiation. After sufficient time, this cooling will rob the hot gas of its internal energy. As the temperature falls in the downstream flow, the density will rise in order to maintain approximate pressure equilibrium. Since the rate of radiative cooling (Fig. 7) increases with falling temperature and rising density, the cooling increases rapidly. One may estimate the timescale for cooling from the equation: d 5 nkT D n n .T / (16) dt 2 e where we have taken coefficient 5/2 rather than 3/2 because the cooling takes place at constant pressure rather than at constant density. Equation (16) yields a characteristic cooling time: 5kT tC (17) Œ2n0 .T / where we have set ne D n, assuming pure hydrogen. Figure 9 illustrates the structure of a radiative shock, calculated by integrating the hydrodynamic equations downstream from the shock front including NIE radiative cooling. The passage of the shock front heats the ions to a temperature T 2 106 K. Initially, the electrons are cooler than the ions, but in a relatively short time Coulomb collisions equilibrate the ion and electron temperatures. In the ionization 3 zone the gas is under-ionized, but by t 10 =n0 years the system has relaxed 4 to CIE. Then, by t 10 =n0 years, radiative cooling causes the temperature to 20 R. McCray Fig. 9 Temperature (thin line) and density (thick line) structure for electrons (dashed line)and protons (solid line) for a radiative shock of velocity 250 km s 1 (Pun et al. 2002) decrease by a factor 10 2, while the density increases by a factor 102. The dense gas in the photoionized zone would continue to cool, but the radiative cooling is balanced by heating due to photoionization by EUV and soft X-rays produced in the cooling region. In this way, roughly half of the radiation produced in the cooling zone is trapped in the photoionization zone and converted to optical radiation, while the other half propagates upstream into the unshocked gas. The photoionization zone radiates optical radiation with a spectrum similar to that of a planetary nebula. Actually, radiative shocks are violently unstable (Pun et al. 2002). With a constant driving pressure, the shock front does not have constant velocity. As radiative cooling sets in, the shock front slows down, causing a drop in temperature of the shocked gas. With decreasing temperature, the shock front slows down more, resulting in a runaway collapse of the cooling layer. The collapse is halted when the photoionization zone catches up with the shock and causes it to speed up again. This instability manifests itself in three dimensions, causing the optical line emission to have a velocity dispersion comparable to the shock velocity. 9.3 Balmer-Dominated Shocks: The Reverse Shock As the name implies, Balmer-dominated shocks are characterized by the emission of strong Balmer lines of hydrogen and little else. Balmer-dominated shocks were first recognized in the spectra of supernova remnants such as Tycho and were interpreted by Chevalier et al. (1980) as the result of the passage of the supernova blast wave into interstellar gas containing neutral hydrogen atoms. Downstream The Physics of Supernova 1987A 21 from the shock, the atoms can be excited and/or ionized by collisions with hot ions. If they are excited to levels n 3, the atoms will radiate Balmer lines. The excited hydrogen atoms retain the velocity distribution that the neutral atoms had before shock passage. Once ionized, the atoms will cease emitting line radiation. On average, 0.2 H˛ photons are emitted for every HI atom that passes through a fast shock. Note that the mechanism described above is entirely different from the mech- anism for Balmer line emission from gaseous nebulae. The former mechanism, electron-ion impact excitation by fast electrons and ions, excites all ions with comparable cross sections, so that the strengths of emission lines from different elements are roughly proportional to the element abundances. Thus, for shocks in gas with Magellanic Cloud abundances, hydrogen emission lines are some four orders of magnitude brighter than emission lines from other elements. In contrast, emission lines from gaseous nebulae are produced by collisions of ions with thermal electrons, which have insufficient energies to excite hydrogen atoms to states n 2. In photoionized nebulae, Balmer lines are produced by radiative recombination, which proceeds at a rate much slower than impact excitation. Moreover, forbidden lines such as [O I] 6300; 6364 and [N II] 6548; 6583 can have strengths comparable to H˛, despite the fact that oxygen and nitrogen have much lower abundance than hydrogen. The Balmer emission from the reverse shock in SN1987A takes place in a different frame of reference from the Balmer emission seen in supernova remnants. In the latter, the blast wave overtakes nearly stationary interstellar gas containing HI atoms, so the Balmer lines have narrow profiles (FWHM 20 km s 1.In addition to this narrow line emission, line profiles from Balmer-dominated shocks in supernova remnants have components with linewidths comparable to the thermal velocity of protons in the shocked gas, caused by charge transfer collisions of hydrogen atoms with hot protons in the shocked plasma. But the cross sections for such charge transfer collisions are unimportant in the high velocity flow through the reverse shock of SN1987A.) In the case of SN1987A, the HI atoms have the velocity of the freely expanding supernova debris, so the Balmer line emission will have Doppler shifts ranging up to several thousand km/s. 9.4 Doppler Tomography As described in Sect. 2, the supernova blast wave propagated through the envelope of the progenitor in a few hours. After that, the debris is further accelerated by the pressure of trapped radiation resulting from the initial explosion and from the depo- sition of energy by the decay of 56Ni (Li et al. 1993). Within a time t 10 days, this acceleration becomes negligible, and the supernova debris expands freely. If we can image the supernova debris at a given Doppler shift within an emission line, we are seeing a slice of the debris at a depth z D vDt. This relation between depth and Doppler shift for supernova debris is a kind of Hubble’s law, vD D H0z, where the Hubble constant H0 D 1=t. In fact, it is more accurate than the Hubble’s 22 R. McCray law that describes the expansion of the universe, which has significant departures due to cosmic evolution and gravitational interactions among galaxies and clusters. The slices in SN1987A are nearly planar: the departures from flatness are ız=z t=t 10 days=104 days 10 3. The slices have finite thickness owing to the molecular velocity dispersion. The fractional thickness is given by z=z cs=vD 3 1 10 for thermal velocity cs 1 km s (assuming temperature 100 K) and Doppler 1 velocity vD 1000 km s . Doppler tomography gives us a unique opportunity to map the three-dimensional structure of the reverse shock in SN1987A with the Space Telescope Imaging Spectrograph (STIS). Figure 10 shows an image of a STIS spectrum with the slit placed along the minor axis of the equatorial ring. Each point in the STIS spectra is located by two parameters: , the angle along the slit, and , the Doppler shift. The physical height of the point is given by h D D , where D 50 kpc is the distance to the supernova, and the projected distance from the midplane of the supernova is given by d D ct = 0.Given h, d, and i D 45ı, the inclination of the equatorial ring, we can remap the STIS image of Fig. 10 into cylindrical coordinates z; , where z is the height above the plane defined by the equatorial ring and is the cylinder radius, as illustrated in Fig. 11. In this way, we transform the STIS image shown in Fig. 10 into a map of the reverse shock shown in Fig. 12. Note that the reverse shock has moved into the equatorial ring, as illustrated in Fig. 5. Fig. 10 STIS spectrum of the reverse shock of SN1987A. (France et al. 2011). Left panel: location of the STIS slit on image of SN1987A. Right panel: STIS spectrum. RS denotes emission from reverse shock. D denotes emission from interior debris. The vertical bar is due to H˛ and [NII] 6548, 6563 emission from nearly stationary gas in the ring, while the pair of spots near the left border is due to emission by [OI] 6364 by the ring Fig. 11 Transformation of coordinates inferred from STIS spectra (d, h) to R cylindrical coordinates ( ,z) ρ z r h i α d The Physics of Supernova 1987A 23 Fig. 12 Map of reverse shock and supernova debris inferred from STIS spectrum (France et al. 2011). The solid red dot denotes the center of the supernova. The red circles denote the equatorial ring. The part of the STIS image along the slit, which represents stationary gas, has been masked out 10 Interior Debris The faint irregular feature near the center of SN1987A seen in Fig. 1 represents H˛ emission from the freely expanding supernova debris. This emission is also evident in the STIS spectrum (the feature marked D in Figs. 10 and 12). Like the reverse shock, this emission can be mapped by Doppler tomography. Figure 12 shows that it is concentrated in the equatorial plane and is predominately blueshifted, with radial velocity extending to 6000 km/s, well beyond the expansion velocity 1800 km/s that was inferred from the profiles of the emission lines during the nebular phase. Note that there is little or no redshifted counterpart to the feature marked D in Fig. 12. Its absence is probably due to extinction by internal dust. Figure 2 shows that the optical luminosity of SN1987A began to increase after 5000 days. Before that time, the optical emission from the debris was more centrally concentrated than the source shown in Fig. 2, and it was fading. The brightening can be attributed to illumination of the debris by the annular source of X-rays caused by the impact of the outer debris with the circumstellar ring (Larsson et al. 2011). The X-rays cannot penetrate into the inner debris where the nucleosynthesis products reside, because the photoabsorption cross sections of these elements are much greater than those of hydrogen and helium. 10.1 Molecular Emission from Inner Debris As the simulations of Fig. 3 show, the structure of the inner debris is sensitive to the mass of the progenitor envelope. Fortunately, it is now becoming possible to map this structure through Doppler tomography of molecular rotational emission lines 24 R. McCray 70 1.0 60 0.8 50 CO3-2 40 0.6 SiO7-6 30 CO2-1 SiO6-5 0.4 SiO5-4 flux density [mjy] 20 SiO8-7 atmospheric transmission − − 0.2 10 0.3 − 0.35 − − 0.35 0.15 0.25 − − HCO+3-2 0.22 0.18 HCO+4-3 0 0.0 220 240 260 280 300 320 340 360 frequency [GHz] Fig. 13 Spectrum of SN1987A as observed by ALMA (Matsuura et al. 2017). The black curve indicates atmospheric transmission at the ALMA site with the Atacama Large Millimeter Array (ALMA). Unlike optical radiation, which is strongly absorbed by the dust in the debris, millimeter radiation can pass through the dust with negligible absorption. Figure 13 shows the spectrum of SN1987A in the mm/sub-mm band as observed with the ALMA. Emission lines from the rotational transitions .j; j 1/ D .2; 1/ and .3; 2/ of CO and (5,4), (6,5), and (7,6) of SiO are clearly evident. (The (8,7) line of SiO is blended with the (3,2) line of CO.) To interpret these results, we need a model for how the observed luminosities of the emission lines depend on the physical conditions in the debris. To characterize the simplest such model we assume that the CO (or SiO) resides in zones all having uniform temperature and density. If so, the luminosity in each emission line from a thin (thickness dz) section displaced by a line-of-sight distance z from the midplane of the debris is given by dLul D h ul Aul nuA.z/dzPesc (18) where Aul is the Einstein A coefficient for the radiative transition u ! l and nu is the number density of molecules in state u. A.z/ is the projected area of a thin section displaced by a line-of-sight distance z from the midplane of the debris, and dz is the thickness of the section. For the freely expanding debris dz D ctd = ul . The mean escape probability, Pesc, accounts for the possibility that a spectral line photon may undergo many resonant scatterings before it escapes the emitting region. In the optically thin limit, Pesc D 1,wehave The Physics of Supernova 1987A 25 Z Lul D h ul Aul nj A.z/dz (19) R where A.z/dz D Vem, the net volume occupied by the emitting gas. Typically, however, a line photon will be reabsorbed and reemitted many times before it escapes. To account for this probability, we calculate the optical depth: Z gl nu ul D dznl lu 1 (20) gunl 2 where lu D . e =mec/flu . / is the absorption cross section, flu is the Roscillator strength for the transition l ! u, and . / is the line profile function, . /d D 1. The term in square brackets accounts for induced emission. Then, taking dz D ctd. /= j;j 1,Eq.(20) becomes 3 2 gl nu 0tgunl Aul gl nu lu D nl . e =mec/flu.ct= ul / 1 D 1 (21) gunl 8 gl gunl The mean escape probability is given by Z 1 1 exp. lu/ Pesc D dx exp. x lu/ D (22) 0 lu In the limit lu 1 Eqs. (19), (20), (21), and (22) yield 4 L D ul B .T /V ; (23) ul ct ul em 3 2 where B .T / D .2h =c /Œ1 exp.h =kTul / is the Planck function and Tul is the excitation temperature, defined by nu=nl D .gu=gl / exp. h ul =kTul /. Note the difference between the expressions for the line luminosity in the optically thin limit (Eq. 19) and the optically thick limit (Eq. 23). In the former case, the luminosity is proportional to nuVem, the total number of molecules in the upper state, while in the latter case, the luminosity is proportional to the net emitting volume, Vem, and is independent of the density of molecules. 10.2 Rotational Transitions of CO and SiO We briefly summarize the physics of rotational transitions of CO and SiO, as described in the NRAO website Molecular Line Spectra (http://www.cv.nrao.edu/ course/astr534/MolecularSpectra.html). The energy levels of the excited rotational states are given by Ej D j.jC1/E0, and the frequency of the transition j ! j 1 is given by 26 R. McCray j;j 1 D 2j B ; (24) where B D 57:65 GHz for CO and B D 21:71 GHz for SiO. Accordingly, we identify the strong emission lines seen in Fig. 13 as the (2,1) transition of CO and the (5,4), (6,5), and (7,6) transitions of SiO. The strong feature at 345 GHz is a blend of the 345.9 GHz (3,2) transition of CO and the 347,4 GHz (8,7) transition of SiO. The Einstein A coefficients for radiative transitions of diatomic molecules are given by 64 4 2j A D 3 ; (25) j;j 1 3hc3 2j C 1 j;j 1 where is the electric dipole moment of the molecule. For CO, D 0:11 Debye D 0:11 10 18 cgs, and for SiO, D 3:15 Debye. It follows that for CO j 4 A D 21:6 10 8 s 1; (26) j;j 1 2j C 1 and for SiO, by j 4 A D 9:15 10 6 s 1: (27) j;j 1 2j C 1 Note that, for a given frequency, the Einstein A coefficient for SiO is 820 times greater than that for CO because the dipole moment of SiO is 28 times greater than that of CO. 10.3 Modeling the Spectral Line Energy Distribution If we have a model for the number densities nj D nf j of molecules in exci- tation states j , we can calculate the luminosities, Lj , of emission lines from Eqs. (19), (20), (21), and (22): Lj;j 1 D h j;j 1Aj;j 1nf j VemPesc.j; j 1/: (28) For example, if the energy levels are populated in local thermodynamic equilibrium (LTE), we have gj fj .Te/ D exp. Ej =kTe/; (29) G.Te/ where gj D 2j C 1, Ej D j.j C 1/h B , and the partition function is, to a good approximation, given by The Physics of Supernova 1987A 27 kT 1 E G.T/ e C C 0 : (30) E0 3 15kTe If the LTE model is valid, one can calculate the luminosity of any given line from Eqs. (21), (22), (23), (24), (25), (26), (27), and (28) by specifying the local density, n, of molecules, the excitation temperature, Te, and the total emitting volume, V . The LTE model will give a good approximation to the level populations, fj , provided that the rate of excitations/deexcitations by collisions exceeds the rate of radiative transitions, i.e., ncollCj;j 1 Aj;j 1Pesc.j; j 1/; (31) ˝ ˛ where ncoll is the density of collision partners and Cj;j 1.Tk/ D v j;j 1 is the rate coefficient for transitions driven by collisions with partners having kinetic temperature Tk. There is considerable uncertainty in the value of ncollCj;j 1. Rate coefficients for excitations of rotational transitions of CO by H2 and He have typical values 11 3 1 C2;1 3:4 10 cm s (Schöler et al. 2005: http://home.strw.leidenuniv.nl/~ moldata/). So, for example, the population of CO in level j D 2 will be in LTE if 4 3 the CO is embedded in molecular hydrogen with n.H2/ 2 10 cm . However, it is unlikely that the CO-emitting gas is predominately H2. If the gas is predominately CO, the relaxation rates for CO C CO collisions may be substantially greater than those for CO C H2 collisions, so the critical density of CO molecules may be much less than that of H2 molecules. To take into account departures from LTE, one can calculate the populations, fj .Tk/, by solving the coupled rate equations: df j D 0 D f R C f R C f R (32) dt j j j C1 j C1;j j 1 j 1;j where Rj D Aj;j 1Pesc.j; j 1/ C ncoll.Cj;j 1 C Cj;jC1/; (33) Rj C1;j D Aj C1;j Pesc.j C 1; j / C ncollCj C1;j (34) Rj 1;j D ncollCj 1;j : (35) Note that the solutions, fj .Tk/, are functions of the kinetic temperature, and Tk,of the colliding partners, in contrast to the rotational excitation temperature, Te,ofthe molecules in the LTE approximation. The code RADEX (http://var.sron.nl/radex/radex_manual.pdf, van der Tak et al. 2007) solves the set of Eq. (32). One can see that the solutions are functions of two local parameters, ncoll and Tk. They also depend on the density, n, of molecules 0 through the escape probabilities Pesc.j; j /, which depend on nj (Eqs. 20 and 22). 28 R. McCray RADEX was developed to interpret observations of molecular line emission from clouds of interstellar gas, characterized by ncoll, Tk, column density, N , and linewidth, v. One can find the solutions by specifying these parameters in the RADEX online site http://var.sron.nl/radex/radex.php. Since RADEX online asks us to specify both N and v, we choose an arbitrary value of v,say, v D 100 km/s, and, for given local density of molecules, n, specify the column density from the relationship N D n vt, where t is the time since explosion. RADEX online returns, for each transition, the excitation temperature defined by f .2j C 1/ h j D exp j (36) fj 1 .2j 1 kTR and the brightness temperature of the emitting surface, TR.j /, defined by 2h 3 h 1 I D B. ;T / D j exp j 1 ; (37) j R 2 c kTR where I .TR/ is the specific intensity radiated by the surface. The luminosity of an emission line is given by Z 4 L D 4 d A.z/I .T / D j B. ;T /V (38) j R ct j R em where we have used the relation d = j D dz=ct. To infer the physical conditions (n; ncoll;TK ; and Vem) of the emitting gas, we search the space defined by these four parameters and map the surface for which the line luminosities predicted by Eq. (38) agree with the observed luminosities. Note that Vem, the volume occupied by the emitting molecules, can be much less than the volume, V , containing the emitting debris. We define a filling factor, f D Vem=V , where V may be estimated from the observed widths and images of the emission lines. If, as is likely, the molecules are confined within unresolved clumps, we expect f<1. The task of inferring the physical conditions (Vem, Tkin, ncoll, and n)fromthe observed lines is still incomplete. Kamenetzky et al. (2013) fitted the luminosities of the CO 2 1 and 1 0 lines observed with ALMA and the CO 7 6 and 6 5 lines observed with the Herschel-SPIRE spectrometer with an LTE model and inferred that the CO-emitting gas had MCO > 0:2Mˇ, Te.CO/ > 100 K. Matsuura et al. (2017) fitted the CO observations with a NLTE RADEX model (assuming ncoll D 5 3 10 cm ) and find MCO D 0:01 0:03Mˇ and Tkin D 30 50 K. Matsuura et al. 5 3 also fitted the SiO 6 5 and 5 4 lines and found MSiO D 2 10 1 10 Mˇ, Tkin D 20 170 K. These estimates of the properties of the emitting regions are based on the ques- tionable assumption that they have uniform densities and temperatures. Indebetouw et al (2017) have employed ALMA to obtain images and spectra of SN1987A at The Physics of Supernova 1987A 29 angular resolution 0:1” When these data become available, we will be able to obtain more refined measurements of the properties of the emitting regions. 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