Physics of Space Elevators: (2) Cable Section:Historically, the main technical problem has been considered the ability of the cable to hold up, with tension, the weight of itself below any particular point. The vertical point with the greatest tension on a space elevator cable is at the level of geostationary orbit, 35,786 km (22,236 mi) above the Earth's equator. This means that the cable material combined with its design must be strong enough to hold up the weight of its own mass from the surface up to 35,786 km. By making any cable larger in cross section at this level compared to at the surface, it can better hold up a longer length of itself. For a space elevator cable, an important design factor in addition to the material is how the cross section area tapers down from the maximum at 35,786 km to the minimum at the surface. To maximize the usable excess strength for a given amount of cable material, the cable's cross section area will need to be designed in such a way that at any given point, it is proportional to the force it has to withstand. This section may contain original research. Please improve it by verifying the claims made and adding inline citations. Statements consisting only of original research may be removed. (February 2012) For such an idealized design without climbers attached, without thickening at high space-junk altitudes, etc. , the cross-section will follow this differential equation: or where g is the acceleration along the radius (m·s−2), S is the cross-section area of the cable at any given point r, (m2) and dS its variation (m2 as well), ρ is the density of the material used for the cable (kg·m−3). σ is the stress the cross-section area can bear without yielding (N·m−2=kg·m−1·s−2), its elastic limit. The value of g is given by the first equation, which yields:, the variation being taken between r1 (geostationary) and r0 (ground). It turns out that between these two points, this quantity can be expressed simply as:, or where is the ratio between the centrifugal force on the equator and the gravitational force