  The temperature in most parts of a protoplanetary disk is determined by
irradiation from the central star. Numerical experiments of Watanabe \& Lin
(2008) suggested that such disks, also called `passive disks', suffer from a
thermal instability. Here, we use analytical and numerical tools to elucidate
the nature of this instability. We find that it is related to the flaring of
the optical surface, the layer at which starlight is intercepted by the disk.
Whenever a disk annulus is perturbed thermally and acquires a larger scale
height, disk flaring becomes steeper in the inner part, and flatter in the
outer part. Starlight now shines more overhead for the inner part and so can
penetrate into deeper layers; conversely, it is absorbed more shallowly in the
outer part. These geometric changes allow the annulus to intercept more
starlight, and the perturbation grows. We call this the irradiation
instability. It requires only ingredients that are known to exist in realistic
disks, and operates best in parts that are both optically thick and
geometrically thin (inside 30AU, but can extend to further reaches when, e.g.,
dust settling is considered). An unstable disk develops travelling thermal
waves that reach order-unity in amplitude. In thermal radiation, such a disk
should appear as a series of bright rings interleaved with dark shadowed gaps,
while in scattered light it resembles a moving staircase. Depending on the gas
and dust responses, this instability could lead to a wide range of
consequences, such as {\w ALMA rings and gaps,} dust traps, vertical
circulation, vortices and turbulence.
