We present results from Spitzer Space Telescope observations of the mid-infrared phase variations of three short-period extrasolar planetary systems: HD 209458, HD 179949 and 51 Peg. We gathered IRAC images in multiple wavebands at eight phases of each planets orbit. We find the uncertainty in relative photometry from one epoch to the next to be significantly larger than the photon counting error at 3.6 micron and 4.5 micron. We are able to place 2-sigma upper limits of only 2% on the phase variations at these wavelengths. At 8 micron the epoch-to-epoch systematic uncertainty is comparable to the photon counting noise and we detect a phase function for HD 179949 which is in phase with the planets orbit and with a relative peak-to-trough amplitude of 0.00141(33). Assuming that HD 179949b has a radius R_J < R_p < 1.2R_J, it must recirculate less than 21% of incident stellar energy to its night side at the 1-sigma level (where 50% signifies full recirculation). If the planet has a small Bond albedo, it must have a mass less than 2.4 M_J (1-sigma). We do not detect phase variations for the other two systems but we do place the following 2-sigma upper limits: 0.0007 for 51 Peg, and 0.0015 for HD 209458. Due to its edge-on configuration, the upper limit for HD 209458 translates, with appropriate assumptions about Bond albedo, into a lower limit on the recirculation occuring in the planets atmosphere. HD 209458b must recirculate at least 32% of incident stellar energy to its night side, at the 1-sigma level, which is consistent with other constraints on recirculation from the depth of secondary eclipse depth at 8 micron and the low optical albedo. These data indicate that different Hot Jupiter planets may experience different recirculation efficiencies.
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