time, Tm, that is, Tc=(Tm)p, for times in seconds. Typical values of p are 1.25 to 1.45, but some are low as 1.1 and high as 1.8. The Kron–Halm catenary equation Kron's equation as modified by Halm states that the response of the film is a function of , with the factor defined by a catenary (hyperbolic cosine) equation accounting for reciprocity failure at both very high and very low intensities: where I0 is the photographic material's optimum intensity level and a is a constant that characterizes the material's reciprocity failure. Quantum reciprocity-failure model Modern models of reciprocity failure incorporate an exponential function, as opposed to power law, dependence on time or intensity at long exposure times or low intensities, based on the distribution of interquantic times (times between photon absorptions in a grain) and the temperature-dependent lifetimes of the intermediate states of the partially exposed grains. Baines and Bomback explain the "low intensity inefficiency" this way: Astrophotography Reciprocity failure is an important effect in the field of film-based astrophotography. Deep-sky objects such as galaxies and nebulae are often so faint that they are not visible to the un-aided eye. To make matters worse, many objects' spectra do not line up with the film emulsion's sensitivity curves. Many of these targets are small and require long focal lengths, which can push the focal ratio far above 5. Combined, these parameters make these targets extremely difficult to capture with film; exposures from 30 minutes to well over an hour are typical. As a typical example, capturing an image of the Andromeda Galaxy at 4 will take about 30 minutes; to get the same density at 8 would require an exposure of about 200 minutes. When a telescope is tracking an object, every minute is difficult; therefore, reciprocity failure is one of the biggest motivations for astronomers to switch to digital imaging. Electronic image sensors have their own limitation at long exposure time and low illuminance levels, not usually referred to as reciprocity failure, namely noise from dark current, but this effect can be controlled by cooling the sensor. Holography A similar problem exists in holography. The total energy required when exposing holographic film using a continuous wave laser (i.e. for several seconds) is significantly less than the total energy required when exposing holographic film using a pulsed laser (i.e. around 20–40 nanoseconds) due to a reciprocity failure. It can also be