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Most launch vehicles define an upper limit on the payload they may carry. For instance - Falcon 9, prepped for launch, has a defined max. payload capability of 13,150 kg.

I would like to know the minimum weight above the rated max. payload capability which would be likely to cause a failure/to stop it even getting off the ground.

By how much may the maximum payload vary without causing a launch failure? What would happen if a launch was attempted with extra mass on board?

This is purely hypothetical, so no need to delve into the reasons why there was extra mass on board. The question is also in a general context and not specific to the Falcon.

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Normally, rocket manufacturers show capability as "payload to LEO" or "payload to GTO". These aren't super-precise; LEO means roughly between 160 and 2000km altitude, which is a pretty broad range. If a rocket can deliver 13,000kg to the 400km altitude of the ISS, it can deliver slightly more than that to 200km, and slightly less than that to 2000km.

Weather conditions and manufacturing variations in individual rocket engines means there has to be a few percent of margin in the specified payload.

The flight control systems know their altitude, speed, and so on, and continually compensate for it; if the rocket is accelerating 0.1% less quickly than it's supposed to because of a 10kg stowaway, it may be able to compensate by firing the orbital insertion stage for a bit longer.

Because of uncertainties in the flight profile (weather, hardware variation, imperfect timing of engine cutoff) it's common for the actual orbit reached to be slightly different from the one intended. This is not a big deal in most cases, and if positioning is critical, the payload will have thrusters of its own to fine-tune its trajectory.

Once you're talking about a human-sized stowaway, that's about 1% of Falcon 9's payload. That would probably make enough difference that the mission controllers would actually notice, but I'd be shocked if there wasn't at least sufficient margin of fuel tankage to handle that -- I'd call it negligence otherwise.

Typically these rockets have an thrust-to-weight ratio at launch of 1.15 or above. For Falcon 9, you'd need 75 tons of additional payload to keep it from leaving the pad entirely. Even then, once you'd burned off some fuel, the rocket would conceivably fly -- except you'd destroy the launch pad first, and that would doubtless destroy the rocket as well.

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Margins vary from launch vehicle to launch vehicle and from mission to mission, but to put a lower bound on it, the final dry mass of the upper stage + payload is often not even known to better than about 0.5%.

For a typical flight (say 5000 kg to LEO), extra (or missing) mass would have to be several tens of kilograms to even be noticeable in the telemetry / trajectory corrections.

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Like Tildal says, this question doesn't really make sense in the real world, as the flight profile depends on the precise mass.

If the payload changed, the launch would be aborted.

If it went undetected before launch, the extra payload would be rapidly detected as it would immediately affect the flight profile. If too heavy, bringing the Thrust/Weight ratio below 1, then it wouldn't even launch, but in any case there would be a measurable effect.

Whether or not it impacted the delivery of payload to the correct orbit would depend on propellant tolerances. It may impact the lifespan of the satellite, if it had to use extra propellant to maneouvre to a correct orbit, etc.

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    $\begingroup$ Would an addition of only a kilo or something have that much of an effect? Enough to make a rocket not launch? $\endgroup$ – Phizzy Jan 19 '15 at 16:32
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    $\begingroup$ Likely not prevent the launch, but definitely impact the flight profile. $\endgroup$ – Rory Alsop Jan 19 '15 at 16:38

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