2.1 Event decoherence time

A consideration in a survey for slow transients is the desire to avoid repeatedly redetecting the same long-duration events. We introduce the concept of the event decoherence function, which characterises how the particular transient events that are detected in a given survey field are correlated in time with those detected in a subsequent observationFootnote ¹ . This quantity then determines the timescale over which successive visits to a given patch of sky would detect a fundamentally new set of transient sources. Where redetection is deemed desirable, the event decoherence function, if known, allows one to set the reobservation time according to the fraction of events to be redetected.

We represent the occurrence of an event by a function f(t; ΔT) such that, when the event occurs f assumes the value one, and it is zero otherwise. For an event that commences at t = 0 and has duration ΔT one has f(t) = H(t) − H(t − ΔT), where H(t) is the Heaviside function. One may use this function to count the number of events occurring over a time interval ${\cal T}$ from the superset containing all ${\cal N}$ events. For events due to a large ensemble of sources with event times t_i and durations ΔT_i , the number of transient events occurring at a time t is given by

(1) \begin{eqnarray} N(t) = \sum _i^{\cal N} f(t-t_i; \Delta T_i). \end{eqnarray}

As we are interested in the correlations between different events, the quantity of interest here is the autocorrelation in the fluctuation in the event count, $\delta N(t) = N(t) - \bar{N}$ . For events which are distributed randomly and independently in time, this is given by (see Appendix A),

(2) \begin{eqnarray} \langle \delta N(t+t^{\prime }) \delta N(t) \rangle &=& \left[ \frac{1}{2 {\cal T}} \sum _j^{\cal N} ( |t+\Delta T_j | + |t-\Delta T_j | - 2 |t| ) \right]\nonumber\\ && -\, {\cal O} \left( \frac{{\cal N} \langle \Delta T\rangle ^2}{{\cal T}^2} \right), \end{eqnarray}

where we henceforth neglect the terms that are second order in ${\cal T}^{-1}$ or higher. We may satisfy ourselves that the formalism introduced here, in terms of event counting, is truly a measure of the timescale over which the detection of a set of events in a field becomes statistically independent from a future measurement. An alternative conceptual approach is to consider the correlation in time between each event and every other event, and then average over all correlations of event pairs. This is equivalent to an average over all pairs of events, and it is evident (see Equation (A1)) that this is mathematically identical to the approach adopted here.

In order to evaluate the decoherence function explicitly we specify a probability distribution for the event durations, ΔT_i and take the continuum limit so that the sum over events becomes an integral. Let us denote p(ΔT)dΔT as the probability of obtaining events between durations ΔT and ΔT + dΔT. Then the average event decoherence time is represented in the form,

(3) \begin{eqnarray} \langle \delta N(t\hspace*{-.5pt}+\hspace*{-.5pt}t^{\prime }) \delta N(t) \rangle &\hspace*{-.5pt}=\hspace*{-.5pt}& \left[ \frac{1}{\cal T} \int p(\Delta T) \left[ (t^{\prime }\hspace*{-.5pt}+\hspace*{-.5pt}\Delta T) H(\hspace*{-.5pt}-\hspace*{-.5pt}t^{\prime }) H(t^{\prime }\hspace*{-.5pt}+\hspace*{-.5pt}\Delta T)\right.\right.\nonumber\\ &&\left.\left. +\, (\Delta T - t^{\prime }) H(t^{\prime }) H(\Delta T - t^{\prime }) \right] d\Delta T \right]. \end{eqnarray}

To acquire an intuitive appreciation of the event decoherence timescale, we evaluate the event decoherence function for three representative cases: events all possessing identical durations, events whose durations follow a gaussian distribution, and event durations that follow a power law distribution.

2.1.1 Single event duration

For a single event duration one has p(ΔT) = δ(ΔT − ΔT ₀), where ΔT ₀ is the duration of all the events. In this case Equation (3) becomes

(4) \begin{eqnarray} \langle \delta N(t+t^{\prime }) \delta N(t) \rangle = \frac{{\cal N}}{2 {\cal T}} \left( |t+\Delta T_0 | + |t-\Delta T_0 | - 2 |t| \right). \end{eqnarray}

This illustrative result shows that, as intuitively expected, the correlation between events is a maximum at t′ = 0 and falls to zero on a timescale of precisely ΔT ₀, the event duration. The amplitude of the autocorrelation is linearly proportional to the event rate, ${\cal N}/{\cal T}$ .

2.1.2 Gaussian duration distribution

For a gaussian distribution of event durations with mean ΔT ₀ and standard deviation σ_ΔT ,

(5) \begin{eqnarray} p(\Delta T) = \frac{1}{\sqrt{2 \pi \sigma _{\Delta T}^2}} \exp \left[ -\frac{(\Delta T-\Delta T_0)^2}{2 \sigma _{\Delta T}^2} \right], \end{eqnarray}

one has

(6) \begin{eqnarray} &&\langle \delta N(t+t^{\prime }) \delta N(t) \rangle \nonumber\\ &&\quad= \frac{{\cal N}}{{\cal T}} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}G(\Delta T_0 - t^{\prime }) (\Delta T_0 - t^{\prime }) + \sigma _{\Delta T}^2 g(\Delta T_0 - t^{\prime }), & t^{\prime } > 0, \\[6pt] G(\Delta T_0 + t^{\prime }) (\Delta T_0 + t^{\prime }) +\sigma _{\Delta T}^2 g(\Delta T_0 + t^{\prime }), & t^{\prime } < 0, \\ \end{array}\right. \nonumber\\ \end{eqnarray}

where

(7) \begin{eqnarray} G(t) &=& \frac{1}{2} \left[1 + {\rm erf} \left( \frac{t}{\sigma _{\Delta T} \sqrt{2}} \right) \right] \quad \hbox{and}\nonumber\\ g(t) &=& \frac{1}{\sqrt{2 \pi \sigma _{\Delta T}^2}} \exp \left[- \frac{t^2}{2 \sigma _{\Delta T}^2} \right] \end{eqnarray}

are the cumulative and probability density functions of a gaussian distribution respectively.

The width of this autocorrelation is determined by both ΔT ₀ and σ_ΔT . When ΔT ₀ is large compared to σ_ΔT , the second term in Equation (6) is small, and the behaviour of the autocorrelation is dominated by the first term. For σ_ΔT small, the function in the first term, $1-{\rm erf}[(t^{\prime }-\Delta T_0)/(\sigma _{\rm \Delta T} \sqrt{2})]$ , behaves like 2H(t′ − ΔT ₀). The result is an autocorrelation function which is dominated by a linear decrease with t′ until the autocorrelation reaches zero at t′ = ΔT ₀.

As σ _T becomes larger one finds that the second term in Equation (6) increases in magnitude, and its effect is to add gaussian wings to the autocorrelation at t′ > ΔT ₀. We note, however that, for the purposes of this particular distribution, it is unphysical to consider situations in which σ _T > ΔT ₀ because the probability distribution of event durations would then imply that a large number of events with negative durations occur.

The event decoherence function declines linearly for lags t′ ≲ ΔT ₀ and then declines exponentially quickly to zero for lags t′ ≳ ΔT ₀.

2.1.3 Power-law event duration distribution

Another likely possibility is one in which the distribution of event durations follows a power law of the form,

(8) \begin{eqnarray} p(\Delta T) = k \Delta T^{-\gamma }, \quad k=\frac{1-\gamma }{\Delta T_{\rm max}^{1-\gamma } - \Delta T_{\rm min}^{1-\gamma }} \hbox{ for }\quad \gamma \ne 1, \end{eqnarray}

for durations between ΔT _min and ΔT _max. For such an event duration distribution, the event decoherence function takes the form,

(9) \begin{eqnarray} \langle \delta N(t+t^{\prime }) \delta N(t) \rangle = \frac{{\cal N}}{{\cal T}} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}F(\Delta T_{\rm min}) - t^{\prime }, & 0< t^{\prime } < \Delta T_{\rm min}, \\ F(t^{\prime }) - t^{\prime } f(t^{\prime }), & \Delta T_{\rm min} < t^{\prime } < \Delta T_{\rm max}, \\ 0, & t^{\prime } > \Delta T_{\rm max}, \\ \end{array}\right. \nonumber\\ \end{eqnarray}

where we define

(10) \begin{eqnarray} F(t) &=& \frac{\gamma -1}{\gamma -2}\left[ \frac{\Delta T_{\rm max}^{2-\gamma } - {t}^{2-\gamma } }{\Delta T_{\rm max}^{1-\gamma } - \Delta T_{\rm min}^{1-\gamma }} \right] \quad \hbox{and} \nonumber\\ f(t) &=& \left[ \frac{\Delta T_{\rm max}^{1-\gamma }-{t}^{1-\gamma }}{\Delta T_{\rm max}^{1-\gamma }-\Delta T_{\rm min}^{1-\gamma }} \right]. \end{eqnarray}

It is obvious that this autocorrelation falls to zero on a timescale t′ = ΔT _max. However, as the value of γ increases, more of the event durations are close to ΔT _min, and the decoherence function is more sharply peaked about the origin; accordingly, the half power point of the function moves to progressively smaller time lags and approaches ΔT _min.

Illustrative plots of the event decoherence function for the various duration probability distribution functions examined here are shown in Figure 1. Hereafter throughout the text we denote T _decoher as the timescale at which the event decoherence function effectively decays to a negligible value.

Figure 1. The decoherence function for three possible event duration distributions, normalised by the event rate $N/{\cal T}$ . The plots show the decoherence function, from left to right, for (i) a set of events all with the same duration ΔT = 1, (ii) a Gaussian duration distribution with ΔT ₀ = 1 and σ_ΔT = 0.4, and (iii) a power law duration distribution with ΔT _min = 0.1, ΔT _max = 1 and γ = 1.2.

2.2 Event types

An important consideration in the optimization of a transients survey relates to the types of events one should optimise for. Our analysis is informed by two premises:

1. i The average event rate is uniform across the survey region on the sky and is homogeneous in time.

2. ii The events that we are interested in optimizing the survey for occur on a duration longer than the duration that the telescope dwells on any one field. This tenet relates to our definition of a slow transient; if the event duration is shorter than the telescope dwell time, no advantage is gained in event rate by slewing the telescope to survey another location. Based on tenet (i), the event detection rate for these faster-timescale transients is identical whether one continues surveying the same patch of sky or moves to another field. Thus the only events worth considering in an optimization of dwell time versus survey area in a slow transients are those whose duration exceeds the dwell time.

Before determining how to optimise the event detection rate within a survey, it is first necessary to ascertain the rate at which a telescope with a given field of view, Ω, and sensitivity would detect transients in a single integration on a single field. We consider a survey with a telescope whose limiting detection flux density within an observing duration T _obs is,

(11) \begin{eqnarray} S_0 = \frac{m k_B T_{\rm sys}}{A_e \sqrt{n_p \Delta \nu T_{\rm obs}}}, \end{eqnarray}

where Δν is the observing bandwidth, A_e the telescope effective area, T _sys the system temperature, n_p the number of polarizations recorded and m the minimum signal-to-noise ratio that is acceptable for detection (e.g. m = 7 for a 7σ detection)Footnote ² .

Some discussion regarding point (ii) above is in order. In practice, the survey will detect events of durations both shorter and longer than the telescope dwell time. For those events with durations whose spans smaller than T _obs, the sensitivity to an event of duration t _tr is smaller than S ₀ by a factor $\sqrt{T_{\rm obs}/t_{\rm tr}}$ , and the detection rate of these “fast” transients will depend in detail on the bi-variate distribution of event luminosity and duration relative to the telescope limiting flux density (which will vary for each event depending on its duration). Thus the survey detection rate for such events is a complicated function of the particular event luminosity-duration distribution that pertains to any given transient population, which is hard to predict a priori. Fortunately, however, by tenet (ii) there is no advantage to gained by trading integration time versus survey area for such transient events. An important point is that the survey will detect just as many of these fast transients by observing the same field as it would by observing another field. As such events make no contribution to the survey rate optimization, we do not consider the contributions of events t _tr < T _obs, further in the present analysis, and our detection rate does not take them into account. The reader is referred to Macquart (Reference Macquart2011) for an analysis of their contribution to the event rate.

In this work we consider events whose luminosity function follows a power law with index − α between luminosities L _min and L _max such that the volume density of objects between luminosities L and L + dL is

(12) \begin{eqnarray} \rho _L dL = \frac{\rho _0 dL}{K} \left\lbrace \begin{array}{l@{\quad}l}L^{-\alpha }, & L_{\rm min} < L < L_{\rm max}, \\[6pt] 0, & \hbox{otherwise}, \end{array} \right. \end{eqnarray}

where ρ₀ is the total event rate per unit volume integrated over all luminosities and the normalization constant is

(13) \begin{eqnarray} K = \left\lbrace \begin{array}{l@{\quad}l}\frac{1}{1- \alpha } \left( L_{\rm max}^{1-\alpha } - L_{\rm min}^{1-\alpha } \right) , & \alpha > 0 \hbox{ and } \alpha \ne 1, \\[8pt] \ln \left( \frac{L_{\rm max}}{L_{\rm min}} \right) & \alpha =1. \\ \end{array} \right. \end{eqnarray}

3.1 Homogeneously distributed events

For a population of transients distributed homogeneously throughout space, the telescope is capable of detecting objects of luminosity L out to a luminosity distance $D_{\rm max} = \sqrt{ L/4 \pi S_0}$ , so that the observed number of events per second for objects of of luminosities between L and L + dL is,

(14) \begin{eqnarray} {\cal R} dL = \rho _L dL \frac{\Omega }{4 \pi } \frac{4}{3} \pi D_{\rm max}^3, \end{eqnarray}

where Ω is the telescope field of view. The total event rate integrated over all luminosities may be expressed in the compact formFootnote ³ ,

(15) \begin{eqnarray} &{}& R_{\rm total} = A \, \Omega \, T_{\rm obs}^{3/4}, \end{eqnarray}

(16) \begin{eqnarray} \hbox{ where } A &=& \frac{\rho _0 }{3 \sqrt{4 \pi } \beta _0^{3/2}} \frac{1}{4 \pi } \left( \frac{1-\alpha }{\frac{5}{2}-\alpha } \right) \frac{L_{\rm max}^{\frac{5}{2}-\alpha }-L_{\rm min}^{\frac{5}{2}-\alpha }}{L_{\rm max}^{1-\alpha }-L_{\rm min}^{1-\alpha }}, \nonumber\\ \alpha &\ne& \left\lbrace 1, \frac{5}{2} \right\rbrace \hbox{ and } S_0 = \frac{\beta _0}{\sqrt{T_{\rm obs}}}. \end{eqnarray}

It is apparent that the detection rate scales linearly with field of view, but increases more slowly with telescope dwell time. This is the basis of the survey optimization: the event detection rate may be increased by sacrificing telescope dwell time in favour of field of view by make multiple pointings over a given survey time T _obs.

3.1.1 Homogeneously distributed events only detectable beyond a minimum distance

The event rate in a survey over a truly homogeneously-distributed population is always nonzero, however how poor the survey sensitivity, by virtue of the fact that the survey may detect events arbitrarily close to the observer.

In certain cases it is more realistic to suppose events are only detectable beyond a certain minimum distance, D _min. This situation effectively pertains to searches in which a certain minimum sensitivity is required before the survey can detect far enough out into the Universe in order to detect any of the target events.

The detection rate between luminosities L and L + dL is therefore

(17) \begin{eqnarray} R dL = \rho _L dL \frac{\Omega }{4 \pi } \left(\frac{4 \pi }{3} D_{\rm max}^3 - \frac{4 \pi }{3} D_{\rm min}^3 \right), \quad D_{\rm max} > D_{\rm min}. \end{eqnarray}

The requirement D _max > D _min implies an inequality between luminosity, sensitivity and D _min: $\sqrt{L/4 \pi S_0} > D_{\rm min}$ , and the detection rate integrated over all luminosities is

(18) \begin{eqnarray} R &=& \frac{\Omega }{4 \pi } \frac{4 \pi }{3} \frac{\rho _0}{K} \int _{L_{\rm min}}^{L_{\rm max}} L^{- \alpha } \left[ \left( \frac{L}{4 \pi S_0} \right)^{3/2} - D_{\rm min}^3 \right]\nonumber\\ &&\times\, H\left[ L - 4 \pi S_0 D_{\rm min}^2 \right] dL. \end{eqnarray}

The behaviour of the detection rate depends on the survey sensitivity relative to the flux density of the brightest and faintest objects detectable at the inner radius of the survey volume. The detection rate is,

(19) \begin{eqnarray} R = \frac{\Omega \rho _0}{3} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}\frac{\zeta (L_{\rm min})}{(4 \pi S_0)^{3/2}} - D_{\rm min}^3 , & \frac{L_{\rm min}}{4 \pi D_{\rm min}^2} > S_0, \\ \frac{\zeta (L^{\prime })}{(4 \pi S_0)^{3/2}} - \eta (L^{\prime }) D_{\rm min}^3 , & \frac{L_{\rm min}}{4 \pi D_{\rm min}^2} < S_0 < \frac{L_{\rm max}}{4 \pi D_{\rm min}^2}, \\ 0, & \frac{L_{\rm max}}{4 \pi D_{\rm min}^2} < S_0. \\ \end{array}\right. \end{eqnarray}

where it is convenient to define,

(20) \begin{eqnarray} \zeta (L) &=& \left(\frac{1-\alpha }{\frac{5}{2}-\alpha }\right) \left( \frac{L_{\rm max}^{5/2-\alpha } - {L}^{5/2-\alpha } }{L_{\rm max}^{1-\alpha } - L_{\rm min}^{1-\alpha }} \right) \quad \hbox{and}\nonumber\\ \eta (L) &=& \left( \frac{L_{\rm max}^{1-\alpha } - {L}^{1-\alpha } }{L_{\rm max}^{1-\alpha } - L_{\rm min}^{1-\alpha }} \right), \end{eqnarray}

and where we define the lowest luminosity detectable at the inner radius of the survey volume as L′ = 4πS ₀ D ² _min. These results may be understood as follows. If the survey sensitivity is so poor that S ₀ > L _max/4πD ² _min, it is incapable of even detecting events at the minimum distance, and the event rate is zero. If the survey is extremely sensitive, S ₀ < L _min/4πD ² _min, the survey detects every event at the inner radius of the survey volume, and the event rate is the same as that for a survey of homogeneously-distributed events except that there is no contribution from events interior to the volume bounded by the radius D _min. In the intermediate case, L _min < L′ < L _max, the survey only detects that fraction of the events whose luminosities exceed L′ at the inner radius of the survey volume.

3.2 A targeted survey of a system at fixed distance

Another possible survey mode involves targeted observations of a galaxy at a known luminosity distance, D_L . Since all the putative transient events are located at the same distance, we detect all events whose luminosity exceeds L ₀ = 4πS ₀ D ² _L . If the total event rate per unit volume is ρ₀, then the total event rate per unit solid angle, $\rho _\Omega$ is found by integrating over the depth of the object, Δz (assumed small compared to its distance) and over the surface area subtended by the portion of sphere of radius D _comoving that encompasses the telescope field of view. An observation over a solid angle Ω would detect a fraction Ω/4π of these objects over the total surface 4πD ² _comoving and subtend an area ΩD ² _A , where D_A = D _comoving/(1 + z) is the angular diameter distance to the object and D_L = (1 + z)² D_A . Thus the total event rate per solid angle is

(21) \begin{eqnarray} \rho _\Omega = \rho _0 \Delta z D_A^2 \Omega , \end{eqnarray}

and so we identify the rate of events between luminosities L and L + dL as

(22) \begin{eqnarray} \rho _{L,\Omega } dL = \frac{\rho _0 \Delta z D_A^2 \Omega }{K} L^{-\alpha } dL = \frac{\rho _0 \Delta z D_A^2 \Omega (1-\alpha ) }{L_{\rm max}^{1-\alpha } - L_{\rm min}^{1-\alpha }} L^{-\alpha } dL. \end{eqnarray}

The total observed detection rate is therefore

(23) \begin{eqnarray} R_{\rm total} &=& \int _{{\rm max}[L_0,L_{\rm min}]}^{{\rm min}[L_0,L_{\rm max}]} \rho _{L,\Omega } dL\nonumber\\ &=& \rho _0 \Delta z D_A^2 \Omega \left\lbrace \begin{array}{l@{\quad}l} 1, & L_0 < L_{\rm min},\\ \eta (L_0), & L_{\rm min} < L_0 < L_{\rm max}, \\ 0, & L_0 > L_{\rm max}. \\ \end{array} \right. \nonumber\\ \end{eqnarray}

The three cases in the above expression are interpreted to mean: (i) when L ₀ < L _min the sensitivity of the observation is so good that even the faintest event can be detected by our observation, so we detect all possible events and the measured event rate is equal to the intrinsic field event rate of ρ₀ΔzD ² _A Ω; (ii) when L ₀ > L _max, the sensitivity of the observation is so poor that not even the most luminous event can be detected, so the event detection rate is zero; and (iii) for the case L _min < L < L _max, the sensitivity is intermediate to the two preceding cases and we detect only a fraction $\frac{L_{\rm max}^{1-\alpha } - L_0^{1-\alpha }}{L_{\rm max}^{1-\alpha } - L_{\rm min}^{1-\alpha }}$ of all the events that occur. An illustration of the behaviour of R _total as a function of L ₀ is shown in Figure 2.

Figure 2. A schematic illustration of the behaviour of the event detection rate, R _tot as a function of the limiting luminosity, L ₀, to which events can be detected in a survey of a system at a fixed distance. If L ₀ is sufficiently low, we detect all objects in the galaxy, but if L ₀ > L _max we detect no events. This plot shows the event rate for a luminosity function for various values of α.

3.2.1 Correction for the effect of refractive interstellar scintillation

As an illustration of the effects of interstellar scintillation on the results presented in this paper, we consider briefly here how interstellar scintillation alters the foregoing result. Interstellar scintillation can randomly amplify or deamplify the signals of compact transients, and so alter the apparent distribution of event flux densities. In situations where the scintillation timescale is long compared to both the event duration and the duration of the observation, scintillation can bear on the detection rate by amplifying some events that would otherwise be too faint to detect.

For events all located at the same distance, as considered in this subsection, there is a linear relationship between source luminosity and flux density. It is then straightforward to incorporate this scintillation effect by considering an “effective” event luminosity distribution that replaces that given in Equation (22) by making the replacement L′ = aL, where a is the amplification caused by scintillation. The resulting distribution of effective event luminosities is then given by,

(24) \begin{eqnarray} \rho _{L^{\prime }} = \int _{-\infty }^\infty p_a(a) \rho _L \left( \frac{L}{a} \right)\frac{da}{|a|}, \end{eqnarray}

where p_a is the probability distribution of amplifications, ρ _L is the distribution of intrinsic luminosities (viz. Equation (12)), and ρ _L′ is the distribution of effective event luminosities after scintillation is taken into account. For the moderate modulation indices typical of refractive scintillation at centimetre and decametre wavelengths (i.e. m much less than one), the amplification probability distribution takes the form,

(25) \begin{eqnarray} p_a(a) = \frac{1}{\sqrt{2 \pi m^2}} \exp \left[ - \frac{(a-1)^2}{2 m^2} \right]. \end{eqnarray}

We assume that the timescale of both the event and of the observation is short compared to the scintillation timescale; if the observed emission from the transient encompasses several (say n) independent scintillations the modulation index is reduced by a factor of n ^1/2. Noting that ρ _L is non-zero only over the range [L _min, L _max], the effective luminosity function becomes,

(26) \begin{eqnarray} \rho _{L^{\prime }} = \frac{1}{\sqrt{2 \pi m^2}} \int _{L/L_{\rm max}}^{L/L_{\rm min}} \frac{da}{a} \exp \left[ - \frac{(a-1)^2}{2 m^2} \right] \, p_L \left( \frac{L}{a} \right). \end{eqnarray}

We are unable to evaluate this integral in closed form for non-integer values of α, however we may still understand the effects of scintillation qualitatively by considering its solution for integer values. The solution for α = 2 is particularly simple:

(27) \begin{eqnarray} \rho _{L^{\prime }} &=& \frac{\rho _0 \Delta z D_A^2 \Omega L^{-2}}{L_{\rm min}^{-1} - L_{\rm max}^{-1}} \left\lbrace \frac{1}{2} \left[ {\rm erf} \left( \frac{L-L_{\rm min}}{\sqrt{2} \, m L_{\rm min}} \right) - {\rm erf} \left( \frac{ L-L_{\rm max} }{\sqrt{2} \, m L_{\rm max}}\right) \right] \right. \nonumber \\ &&\left. -\, \frac{m}{\sqrt{2 \pi }} \left[ \exp \left( - \frac{( L - L_{\rm min})^2}{2 m^2 L_{\rm min}^2} \right) - \exp \left( - \frac{ (L - L_{\rm max})^2}{2 m^2 L_{\rm max}^2 } \right) \right] \right\rbrace . \nonumber\\ \end{eqnarray}

Scintillation only alters the shape of ρ _L′ near L = L _min and L = L _max; for luminosities in the range $[L_{\rm min} (1+\sqrt{2} m),L_{\rm max} (1-\sqrt{2} m) ]$ the effective luminosity distribution is, to an excellent approximation, ρ _L′ = L ⁻²/(L ⁻¹ _min − L ⁻¹ _max). A plot of the behaviour of this function is shown in Figure 3. It shows that the effect of scintillation is to extend the low and high flux density limits of the distribution to yet lower and higher values respectively. The behaviour of the effective luminosity distribution function near L _min and L _max is dominated by the error functions in Equation (27), from which we deduce that the effect of scintillation is to extend a small fraction of high luminosity events to yet higher luminosities, $\approx L_{\rm max}( 1+ \sqrt{2} m)$ , and some events at the low luminosity end down to values $\approx L_{\rm min} (1- \sqrt{2} m)$ .

Figure 3. The effective luminosity function (blue curve) of Equation (27) for m = 0.1, L _min = 0.01 and L _max = 1. Overplotted in black is the intrinsic luminosity function, ρ _L .

In practice, for the small modulation indices (m < 0.1) typically observed for refractive interstellar scintillation, the effects of scintillation on the event detection rate are small unless the survey sensitivity is such that the limiting luminosity of the observations, L ₀, is close to L _max. We note, however, that there are cases in which the refractive modulation index is observed to be higher ( ≈ 0.3), and that compact objects are also subject to the effects of Extreme Scattering Events (ESEs; Fiedler et al. 1987). ESEs may produce flux density deviations exceeding 50% in some cases, but these events are rare, estimated at 0.013 src⁻¹ year⁻¹.

4.1 Homogeneously distributed objects

Equation (15) presents the event detection rate for a telescope whose sensitivity is such that it can detect objects down to a limiting flux density of S ₀ in a duration T _obs. The event detection rate for a survey over N fields for the same duration is then

(29) \begin{eqnarray} R_{\rm N-fields} = A \, N \Omega \, \left( \frac{T_{\rm obs} - (N-1) t_{\rm slew}}{N} \right)^{3/4}, \end{eqnarray}

and the maximum event rate occurs when the number of fields visited is

(30) \begin{eqnarray} N_{\rm max} = \frac{T_{\rm obs}+ t_{\rm slew}}{4 t_{\rm slew}}. \end{eqnarray}

The associated maximum event rate is

(31) \begin{eqnarray} R_{\rm max} = \frac{3^{3/4} \,A \, \Omega }{4} t_{\rm slew}^{-1/4} \left(T_{\rm obs} + t_{\rm slew} \right), \end{eqnarray}

which is a factor ~ 0.57(T _obs/t _slew)^1/4 larger than the event rate that would be obtained by spending the entire integration time on a single field.

A few obvious points are in order. We have regarded N as a continuous variable, but in practice N is an integer, so it will only be possible to approximately obtain the maximum event rate. Where N _max is less than one, the optimal strategy is to survey only a single field.

Equation (30) indicates that, as T _obs increases, the number of fields that should be searched increases similarly. However, there is a practical limitation to the number of fields to be searched which is set by the physics of the target transient events. After a duration comparable to the event decoherence time one expects each of the searched fields to contain a new set of transient events, so that a second measurement of the same field would be statistically independent of the original measurement. Thus a search of the same field after a duration T _decoher is statistically equivalent to searching another (as yet) unsearched field. As there is no advantage to searching other statistically independent fields over the same fields a second time, this allows us to identify a practical search cycle time as T _decoher. Thus the maximum number of fields that the survey should cycle over is N _max = (T _decoher + t _slew)/4t _slew.

4.1.1 Homogeneously distributed objects detectable only beyond a minimum distance

Here we examine the optimum detection strategy for the slightly more complicated case in which the transient events are only detectable beyond a certain minimum distance. For the sake of algebraic simplicity we assume that the slewing time is negligible relative to observing time, so that the total amount of time spent per field is T _obs/N. In this case the detection rate is obtained from Equation (19) by making the replacement Ω → ΩN and $S_0 \rightarrow S_0 \sqrt{N}$ :

(32) \begin{eqnarray} &&\hspace*{-5pt}R_{\rm N-fields} = \frac{N \Omega \rho _0}{3}\nonumber\\ &&\hspace*{-5pt}\quad\times\,{\!} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}\frac{ \zeta (L_{\rm min}) }{(4 \pi N^{1/2} S_0)^{3/2}} - D_{\rm min}^3 , & \frac{L_{\rm min}}{4 \pi D_{\rm min}^2} > N^{1/2} S_0, \\[6pt] \frac{ \zeta (N^{1/2} L^{\prime }) }{(4 \pi N^{1/2} S_0)^{3/2}} - \eta (N^{1/2} L^{\prime }) D_{\rm min}^3 , & \frac{L_{\rm min}}{4 \pi D_{\rm min}^2} < N^{1/2} S_0 < \frac{L_{\rm max}}{4 \pi D_{\rm min}^2}, \\[6pt] 0, & \frac{L_{\rm max}}{4 \pi D_{\rm min}^2} < N^{1/2} S_0 . \\ \end{array}\right. \nonumber \\ \end{eqnarray}

The behaviour of this function is shown in Figure 4, from which it is clear that the maximum event rate occurs at N > 1 for luminosity functions shallower than α = 3.

Figure 4. A plot of the event detection rate for the case in which the events are distributed homogeneously throughout space, but are only detectable beyond some minimum distance from the observer, D _min. Here we have chosen L _min = 0.001, L _max = 10, S ₀ = 0.05 and D _min = 1. Shown in different colours and on different scales are the event rate plots for α = 0.4 (blue), α = 1.4 (green), α = 2.4 (red) and α = 3.4 (brown). Note that only the curves with α < 3 exhibit a peak as function of N. For larger values of α the detection rate is a monotonically decreasing function of N.

We derive an exact expression for the value of N that optimises the event rate by differentiating Equation (32). For most cases of interest, in which L _max ≫ L _min, the maximum rate occurs in the range L _min/4πS ₀ D ² _min < N ^1/2 < L _max/4πS ₀ D ² _min. The location of the maximum is given by the lowest nonzero solution to the following equation:

(33) \begin{eqnarray} 0&=& 3 (\alpha -3) {L^{\prime }}^{5/2} N^{5/4} + (\alpha -1) {L^{\prime }}^{\alpha } L_{\rm max}^{5/2} N^{\alpha /2}\nonumber\\ && -\, 2 (2 \alpha -5) {L^{\prime }}^{3/2+\alpha } L_{\rm max} N^{3/4+\alpha /2} . \end{eqnarray}

As is obvious from Figure 4, the optimal number of survey fields decreases as the luminosity function becomes progressively steeper. Equation (33) also has the solution N = (L _max/4πS ₀)², which corresponds to the point at which the event detection rate reaches zero; for α > 3 this is the only nonzero solution, reflecting the fact that when α > 3 the detection rate becomes a monotonically decreasing function of N, and the optimal strategy is to survey only a single field.

The turning point may, in principle, instead occur in the regime N ^1/2 < L _min/4πS ₀ D ² _min, in which case it occurs at a value of N given by

(34) \begin{eqnarray} N = \left[ \frac{1}{2 D_{\rm min}^3 (4 \pi S_0)^{3/2}} \left( \frac{1-\alpha }{5/2-\alpha } \right) \frac{L_{\rm max}^{5/2-\alpha } - L_{\rm min}^{5/2-\alpha }}{L_{\rm max}^{1-\alpha } - L_{\rm min}^{1-\alpha }} \right]^2. \end{eqnarray}

Combining this solution with the inequality, N < (L _min/L′)², we see that this solution is only valid when

(35) \begin{eqnarray} \zeta (L_{\rm min}) < 2 L_{\rm min} {L^{\prime }}^{1/2}. \end{eqnarray}

In most practical situations this alternative solution may be ignored.

4.2 A targeted survey of systems at fixed distance

Let us now consider the detection rate over a set of targeted objects at fixed distance. This situation is applicable to a survey of an individual galaxy or cluster of galaxies (e.g. the Virgo supercluster) whose angular size subtends more than one telescope beam, so that multiple pointings are required to cover it. More generally, the situation is applicable to any survey of multiple targets which are approximately equidistant but located in separate fields of view.

We start from Equation (23), which relates the detection rate in a single field to the luminosity distribution of the transient population and our telescope sensitivity, expressed in terms of the limiting detectable luminosity, L ₀ = 4πD ² _L S ₀. Now, since S ₀∝T ^−1/2 _obs, we see that the limiting luminosity of a survey of total duration T _obs which visits N fields is approximately $\sqrt{N} L_0$ . If we account for slewing time, we see that the total time per observation is (T _obs − (N − 1)t _slew)/N, so the limiting luminosity increases to $L_{\cal N} = L_0 \sqrt{N T_{\rm obs}/[T_{\rm obs} - (N-1) t_{\rm slew}]}$ . The detection rate for a survey which visits N fields is then,

(36) \begin{eqnarray} &&\hspace*{-5pt}R_{\rm N-fields} = N \rho _0 \Delta z D_A^2 \Omega \nonumber\\ &&\hspace*{-5pt}\quad\times\,{\!}\left\lbrace \begin{array}{ll}0, & L_0 \sqrt{\frac{N T_{\rm obs}}{T_{\rm obs} - (N-1) t_{\rm slew}}}> L_{\rm max}, \\ \eta \left( L_0 \sqrt{\frac{N T_{\rm obs}}{T_{\rm obs} - (N-1) t_{\rm slew}}} \right) , & L_{\rm min} < L_0 \sqrt{\frac{N T_{\rm obs}}{T_{\rm obs} - (N-1) t_{\rm slew}}} < L_{\rm max}, \\ 1 , & L_0 \sqrt{\frac{N T_{\rm obs}}{T_{\rm obs} - (N-1) t_{\rm slew}}} < L_{\rm min}. \\ \end{array} \right. \nonumber \\ \end{eqnarray}

A plot of the behaviour of this function is shown in Figure 5. An important characteristic of this function is that the detection rate is found to peak for survey luminosities $L_{\rm min} < L_{\cal N} < L_{\rm max}$ for 0 < α < 3. For α > 3, we see that the optimal detection rate occurs at the point where $L_{\cal N} = L_{\rm min}$ . The reason for this relates to the steepness of the luminosity distribution. For distributions steeper than α > 3, most of the events occur at the low luminosity end, and thus a survey that seeks to maximise detection rate must ensure that it achieves a sensitivity sufficient to reach this minimum luminosity in each field that it surveys. For such a distribution, the optimal number of fields corresponds to a strategy in which the time available is subdivided so that each field is surveyed to a sensitivity sufficient to detect objects of L _min (but obviously no lower). Conversely, if the luminosity distribution is shallower than α = 3, there are not enough events at the low luminosity end to justify spending the integration time on that field to reach the sensitivity to detect them. The optimal strategy is to increasingly favour field of view over sensitivity as the luminosity distribution becomes progressively shallower. This point is illustrated in Figure 5, where we see that the peak detection rate occurs at larger N for the case α = 0.5 relative to the case α = 2.5.

Figure 5. The dependence of detection rate on the number of fields surveyed during the interval T _obs. These particular plots show the behaviour of detection rate for (top) α = 0.5, (centre) α = 2.5 and (bottom) α = 3. In the simple case shown here in which t _slew = 0, the break points of the rate curve occur at (L _min/L ₀)² and (L _max/L ₀)². Note that the peak detection rate occurs in the range L ² _min/L ₀ ² < N < L _max/L ² ₀ for 0 < α < 3 but occurs at N < L ² _min/L ₀ ² for α > 3.

For the case 0 < α < 3, we obtain the location and value of the optimal detection rate by finding the turning point of R _N-fields. One has

(37) \begin{eqnarray} &&\hspace*{-6pt}\frac{d R_{\rm N-fields}}{dN} \nonumber\\ &&\hspace*{-11pt}\quad= {\!}\left\lbrace \begin{array}{l}\rho _0 \Delta z D_A^2 \Omega , \\ \rho _0 \Delta z D_A^2 \Omega \left[ \frac{ L_{\rm max}^{1-\alpha } [T_{\rm obs} - (N-1) t_{\rm slew}] + L_{\cal N}^{1-\alpha } \left[ \frac{T_{\rm obs} (\alpha -3)}{2} + t_{\rm slew} \left( N + \frac{\alpha -3}{2} \right) \right] }{ [T_{\rm obs} - (N-1) t_{\rm slew}] [L_{\rm max}^{1-\alpha } - L_{\rm min}^{1-\alpha }] } \right], \\ 0, \\ L_{\cal N} < L_{\rm min},\\ L_{\rm min} < L_{\cal N} < L_{\rm max}, \\ L_{\cal N} > L_{\rm max}. \\ \end{array} \right. \nonumber\\ \end{eqnarray}

For t _slew = 0, the location of the turning point, at dR/dN = 0, has the simple solution:

(38) \begin{eqnarray} N_{\rm max} = \left( \frac{L_{\rm max}}{L_0} \right)^2 \left( \frac{3 -\alpha }{2} \right)^{2/(\alpha -1)}, \end{eqnarray}

whose associated maximum detection rate is

(39) \begin{eqnarray} R_{\rm max} = \Omega \rho _0 \Delta z D_A^2 \left(\frac{1-\alpha }{3-\alpha } \right) \left( \frac{L_{\rm max}}{L_0} \right)^2 \frac{L_{\rm max}^{1-\alpha } }{L_{\rm max}^{1-\alpha } - L_{\rm min}^{1-\alpha } } . \end{eqnarray}

Note that the maximum detection rate depends on L ⁻² ₀, and thus S ⁻² ₀. Thus, in this special instance, the maximum detection rate scales as ΩS ⁻² ₀∝Ω(A_e /T _sys)², which is the metric employed in some telescope survey figures of merit (Cordes 2007, 2009).

More generally, for t _slew ≠ 0, the maximum detection rate occurs at the value of N which satisfies the following transcendental equation:

(40) \begin{eqnarray} \left( \frac{ L_{\rm max} }{ L_0 } \right)^{1-\alpha } &=& \left( \frac{N T_{\rm obs}}{T_{\rm obs} - (N-1) t_{\rm slew}} \right)^{(1-\alpha )/2} \nonumber\\ &&\times\,\left[ \frac{(T_{\rm obs} + t_{\rm slew}) (3 -\alpha ) - 2 N t_{\rm slew} }{ 2[ T_{\rm obs} - (N-1) t_{\rm slew}] } \right]. \end{eqnarray}

As a final remark, we note that it is possible in principle for the value of N _max to exceed the total number of independent telescope pointings, N _point, required to cover the entire survey area. This implies that the optimal strategy includes re-surveying a number of fields for new transients. A re-observation of a previously visited field can be considered statistically independent of a prior observation, and should detect an entirely new set of transient events, as long as the time between observations of the same field exceeds the event decoherence time (see §2.1 above). In the opposite limit, in which the event decoherence time is long compared to the interval between observations of the same field, the assumption underpining the foregoing calculation, namely that each observation of a field is statistically independent, is violated. However, in this case we see trivially that iff N _max > N _point then the optimal strategy is simply to survey each field only once for a duration T _obs/N _point (hence N _max = N _point).