The Toothbrush radio relic associated with the merging cluster 1RXS J060303.3 is presumed to be produced by relativistic electrons accelerated at merger-driven shocks. Since the shock Mach number inferred from the observed radio spectral index, Mradio  2:8, is larger than that estimated from X-ray observations, MX . 1:5, we consider the re-acceleration model in which a weak shock of Ms  1:2 - 1:5 sweeps through the intracluster plasma with a preshock population of relativistic electrons. We nd the models with a power-law momentum spectrum with the slope, s  4:6, and the cutoff Lorentz factor, e;c  7-8104 can reproduce reasonably well the observed proles of radio uxes and integrated radio spectrum of the head portion of the Toothbrush relic. This study conrms the strong connection between the ubiquitous presence of fossil relativistic plasma originated from AGNs and the shock-acceleration model of radio relics in the intracluster medium.

Abstract
1. INTRODUCTION
2. DSA MODEL
    2.1. Electron and Radiation Spectra of Radio Relics
    2.2. Width of Radio Relics
3. NUMERICAL CALCULATIONS
    3.1. DSA Simulations for 1D Spherical Shocks
    3.2. Models for Magnetic Fields
    3.3. Preshock Electron Population
    3.4. Shock Parameters
4. RESULTS OF DSA SIMULATIONS
    4.1. Surface Brightness and Spectral Index Profiles
    4.2. Integrated Spectrum
5. SUMMARY
REFERENCES

• Hyesung Kang(Department of Earth Sciences, Pusan National University) Corresponding author