Dynamic interactions between the two Magellanic Clouds have flung large quantities of gas into the halo of the Milky Way, creating the Magellanic Stream, the Magellanic Bridge, and the Leading Arm (collectively referred to as the Magellanic System). In this third paper of a series studying the Magellanic gas in absorption, we analyze the gas ionization level using a sample of 69 Hubble Space Telescope/Cosmic Origins Spectrograph sightlines that pass through or within 30 degrees of the 21 cm-emitting regions. We find that 81% (56/69) of the sightlines show UV absorption at Magellanic velocities, indicating that the total cross section of the Magellanic System is ~11 000 square degrees, or around a quarter of the entire sky. Using observations of the Si III/Si II ratio together with Cloudy photoionization modeling, we calculate that the total mass (atomic plus ionized) of the Magellanic System is ~2.0 billion solar masses, with the ionized gas contributing over twice as much mass as the atomic gas. This is larger than the current-day interstellar H I mass of both Magellanic Clouds combined, indicating that they have lost most of their initial gas mass. If the gas in the Magellanic System survives to reach the Galactic disk over its inflow time of ~0.5-1.5 Gyr, it will represent an average inflow rate of ~3.7-6.7 solar masses per year, potentially raising the Galactic star formation rate. However, multiple signs of an evaporative interaction with the hot Galactic corona indicate that the Stream may not survive its journey to the disk fully intact, and will instead add material to (and cool) the corona.
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