Lightsails and ionsails [pdf] are not subject to Tsiolkovsky. They are proposed for "Starshot" missions for low-mass cargo toward a 0.1C speed to shoot past another solar-system, or maybe past a Kuiper object. They might also work for long-term course-correction, like Venus / Earth Hohmann - again, if low mass. Or for stationkeeping. This post mainly considers fast-track to deep space.
McGill Interstellar Flight Research Group thinks they can do a Mars-and-back mission with sails. This expands Jeff Greason's idea; it looks like they'd be using magnets as sails, against solar plasma - so, ionic. In related news we just lost the Planetary Society's physical sail around the same time the SLS-launched Near-Earth Asteroid Scout has been lost (according to ToughSF).
Caryn Bailer-Jones a couple years ago floated "The sun diver: Combining solar sails with the Oberth effect". This article talks "photons" so addresses the lightsail. The arithmetical constant she (I assume) proposed was the Light Number λ. This is a ratio between the sail's mass, reflectivity, geometry, and proportions; against our Sun's luminosity (or perhaps a laser-beam's). CBJ noted that as long as λ < 1/2, we have an ellipse; if greater than that, it's a hyperbola to infinity and beyond. Light Sail 2 was 0.01; one starshot proposal would run 0.78. CBJ suspects initial λ might not get that good with a decent cargo, so wants a boost. She's further assuming these missions are unmanned.
With those constraints, CBJ argues for Tsiolkovsky as that booster. If she can get the sail-plus-cargo close to the Sun and running hot (as it were), she can then and there detach the propellant-tank and hoist sail to make the most of low-ish λ. Hey: maybe it can be an R-1.17 ionsail. As for the booster: that's Elon's problem.
Her first proposal was, as the title suggests, Crazy Hermann. UPDATE 1:55 PM - which Elon is going to absolutely soak her for. But she floats a better trajectory.
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