Stellar tidal streams provide an opportunity to study the motion and structure of the disrupting galaxy as well as the gravitational potential of its host. Streams around the Milky Way are especially promising as phase space positions of individual stars will be measured by ongoing or upcoming surveys. Nevertheless, it remains a challenge to accurately assess distances to stars farther than 10 kpc from the Sun, where we have the poorest knowledge of the Galaxys mass distribution. To address this we present observations of 32 candidate RR Lyrae stars in the Orphan tidal stream taken as part of the Spitzer Merger History and Shape of the Galactic Halo (SMHASH) program. The extremely tight correlation between the periods, luminosities, and metallicities of RR Lyrae variable stars in the Spitzer IRAC $mathrm{3.6 mu m}$ band allows the determination of precise distances to individual stars; the median statistical distance uncertainty to each RR Lyrae star is $2.5%$. By fitting orbits in an example potential we obtain an upper limit on the mass of the Milky Way interior to 60 kpc of $mathrm{5.6_{-1.1}^{+1.2}times 10^{11} M_odot}$, bringing estimates based on the Orphan Stream in line with those using other tracers. The SMHASH data also resolve the stream in line--of--sight depth, allowing a new perspective on the internal structure of the disrupted dwarf galaxy. Comparing with N--body models we find that the progenitor had an initial dark halo mass of approximately $mathrm{3.2 times 10^{9} M_odot}$, placing the Orphan Streams progenitor amongst the classical dwarf spheroidals.
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