3.1. Reverberation Mapping Assumptions

There are a number of simplifying assumptions that we can now make:

1. The continuum originates in a single central source. The size of an accretion disk around a supermassive (say, 10^7-8 M) black hole is of the order of 10^13-14 cm. A typical size for the BLR in the same system would be of order a few light days, i.e., ~ 10¹⁶ cm. Note that it is specifically not required that the continuum source emits radiation isotropically, though this is a useful starting point. The "point-source" assumption greatly simplifies the reverberation process. However, we should mention that the point-source assumption is probably not applicable to X-ray reverberation, as the Fe K emission and the hard X-rays that drive this line probably arise in regions that are of similar size, and perhaps co-spatial ⁷⁹.

2. Light-travel time is the most important time scale. We assume specifically that emission-line clouds respond instantaneously to changes in the continuum flux. The time scale to re-establish photoionization equilibrium is the recombination time,

   (17)

   The time it takes a Lyman photon to diffuse outward from the Strömgren depth is about 20 times the direct light-travel time ³⁵,

   (18)

   We also need to carry out our reverberation-mapping experiment on a time scale short enough that the structure of the BLR can be assumed to be stable. The dynamical, or cloud-crossing, time scale for the BLR is typically

   (19)

   where V_FWHM is the Doppler width of the broad line for which the response time = r/c has been measured. Any reverberation-mapping experiment has to be short relative to the crossing time or the structural information might be washed out by cloud motions.

3. There is a simple, though not necessarily linear, relationship between the observed continuum and the ionizing continuum.