We consider radio networks modeled as directed graphs. In ad hoc radio networks, every node knows only its own label and a linear bound on the size of the network but is unaware of the topology of the network, or even of its own neighborhood. The fastest currently known deterministic broadcasting algorithm working for arbitrary n-node ad hoc radio networks, has running time script O sign(n log2 n). Our main result is a broadcasting algorithm working in time script O sign(n log n log D) for arbitrary n-node ad hoc radio networks of eccentricity D. The best currently known lower bound on broadcasting time in ad hoc radio networks is Ω(n log D), hence our algorithm is the first to shrink the gap between bounds on broadcasting time in radio networks of arbitrary eccentricity to a logarithmic factor. We also show a broadcasting algorithm working in time script O sign(n log D) for complete layered n-node ad hoc radio networks of eccentricity D. The latter complexity is optimal.