  We compute the cross section for scattering of light string probes by
randomly excited closed strings. For high energy probes, the cross section
factorizes and can be used to define effective form factors for the excited
targets. These form factors are well defined without the need for infinite
subtractions and contain information about the shape and size of typical
strings. For highly excited strings the elastic form factor can be written in
terms of the `plasma dispersion function', which describes charge screening in
high temperature plasmas.
