The distance to clusters: Correcting for asphericity

X-ray and Sunyaev-Zeldovich (SZ) effect observations can be combined to measure the distance to clusters of galaxies. The Hubble constant, H-0, can be inferred from the distance to low-redshift clusters. With enough clusters to measure the redshift-distance relation out to a redshift z similar to 1, it may be possible to determine the total matter density, 0, and the cosmological constant, Lambda(0), as well. If the intracluster gas distribution is not spherical but elongated by a factor of Z along the line of sight, the inferred distance is increased by Z, and H-0 is decreased by the same factor. Averaging the inferred value of H-0 over a sufficiently large sample of clusters can reduce any systematic bias due to cluster shapes, provided the clusters are selected without any preferred orientation. Even so, elongation contributes significantly to the variance in the measured distances and in the inferred value of H-0. With the addition of gravitational lensing observations, it is possible to infer the three-dimensional shape of an individual cluster, provided the gas is in hydrostatic equilibrium. We demonstrate a specific method for finding the shape and correcting the measured distances to individual clusters. To test this method, we apply it to artificial observations of idealized ellipsoidal model clusters. We base the artificial X-ray observations on the Chandra X-Ray Observatory. For the SZ effect, we assume modest improvements over current observations at the Owens Valley Radio Observatory. We recover the true distances to each of our model clusters without detectable bias and with statistical errors due to measurement uncertainties of 4%-6%.