Here is my understanding, let me know if I've got it right or wrong:

Until recently, most cryogenic propellants were at or close to their boiling points. Each unit of heat leaking into the tank would then boil off a corresponding unit of propellant and the temperature would stay constant. This is roughly like adding near boiling "make-up" water to a pot of boiling water. It might be a little cooler, but not much.

But in the case of sub-cooled LOX, the temperature is way below the boiling point, somewhere between 10 and 30 degrees C depending on the case. When heat leaks into the tank, it will raise the temperature of the LOX which then expands. A full tank would then overflow - remaining full but with less mass of LOX due to the density change.

There would be no boil-off to "make up" by topping it off. It would remain full but steadily decrease in density as it warmed up.

So for a rocket with first stage tank filled with well sub-cooled LOX, why would it need to be "topped-off" until the last minute or so before launch?

This video at T -00:05:05, video time 06:52, my approximate transcription of the narration:

Right now we’re just coming up on the five minute mark here, we are concluding the loading of the RP-1 on to the first stage […] and we are topping-off liquid oxygen as well on that first stage, and we’re going to keep topping that off for about two more minutes.

The video is already queue'd up at the appropriate time code (in this case I think the video edit is final):

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    $\begingroup$ Sure, the LOX gets warmer and less dense. But evaporation is happening as well - compare to a pot of water at 80°C, it won't stay full for a long time. $\endgroup$ – asdfex May 2 '17 at 8:18
  • $\begingroup$ @asdfex very often (usually) in cryogenic systems, a 100% saturated vapor layer exists above the liquid within the cryostat. For example, if you add a simple cover over your 80°C water, or place a lid loosely over it, evaporation loss will cease. $\endgroup$ – uhoh May 2 '17 at 8:59
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    $\begingroup$ there are 2 possible meanings of 'to top off'. 1. finish loading at a lower speed, or 2. make up for volume lost to evaporation. I'm not sure which meaning is used here. $\endgroup$ – Hobbes May 2 '17 at 9:06
  • $\begingroup$ To be precisely, in the case of sub-cooled LOX, the temperature is way below the boiling point ( of LOX at a pressure of 1 bar ). But when the pressure is reduced, the boiling point drops. This is true not only for total pressure, but also for partial pressure of oxygen. $\endgroup$ – Uwe May 2 '17 at 19:20
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    $\begingroup$ My comment was only for the time before launch. The temperature of sub cooled LOX is below the boiling point of LOX at 1 bar. But the process of sub cooling uses the fact the boiling point is lower for the bubbles of helium with a total pressure of 1 bar and a partial pressure of oxygen much lower than 1 bar. If sub cooled LOX is exposed to a lower ambient pressure than 1 bar after launch when the rocket is leaving the upper layers of the atmosphere, it will boil again. If this happens to a second stage while the first stage is still active, some of the oxygen is lost by venting. $\endgroup$ – Uwe May 2 '17 at 20:13

At a pressure of 1 bar, the temperature of liquid boiling oxygen stabilizes at 90 K. For sub-cooling of LOX, the temperature should be lower. It is possible to cool LOX by forced evaporation by a pressure lower than 1 bar. But the LOX tank in a rocket should be as light as possible. If the pressure inside the tank is substantially lower than outside, extra strength and weight is necessary. But according to these papers: (1) (2) and (3) there is another method.

Cold helium gas is injected at the bottom of the tank and the bubbles raise in the LOX. At the surface of the bubbles, LOX evaporates into the bubble and cools the remaining LOX. But extra space is needed for the bubbles in the LOX and for the gas mixture of helium and oxygen above the liquid level. For topping off, the injection of helium is stopped and the remaining space is filled with LOX. Figure 8 of the first paper shows the effect of different helium gas temperatures. The cooling works best with helium at 85 K, but even helium at 150 K cools the LOX.

A bubble injected into the LOX consists of 100 % helium and 0 % oxygen at first. The LOX around this bubble would boil just like in a vaccum because the partial pressure of oxygen in this bubble is zero. Even a bubble consisting of 50 % helium and 50 % oxygen is able to cool LOX at 90 K. Without sub cooling in a tank with boiling LOX at 90 K, the gas above the liquid is 100 % oxygen and the partial pressure of oxygen is 1 bar. If the partial pressure of oxygen is lower than 1 bar in the gas above the liquid or inside the bubbles, the LOX is cooled by evaporation.

At the launch pad the LOX may be precooled using a heat exchanger with ground suplied liquid nitrogen boiling at 77.355 K. To save weight of the rocket, this heat exchanger should be outside the rocket but close to it. Liquid nitrogen and oxygen should not be mixed to avoid solving of nitrogen within the LOX. Cooling with helium bubbles may be used within the rocket LOX tank.

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    $\begingroup$ Fantastic! This is the miracle of stackexchange. I've been wondering about this for a long time, but when I finally remember to ask, bingo! an excellent answer in hours! Thank you. So this is a simple method of in situ refrigeration without complicated moving parts. I think it's quite "cool" (pardon the pun) that hot helium can cool the LOX. It's a little bit like cooling off by sweating in a hot desert breeze. $\endgroup$ – uhoh May 2 '17 at 9:08
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    $\begingroup$ Then you took the time to put it together in an organized fashion and explain it clearly for us and future readers. $\endgroup$ – uhoh May 2 '17 at 9:31
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    $\begingroup$ This helium bubbling trick was also used in the shuttle system prelaunch to make sure that the LOX in the long downcomer line stayed nice and cool. A GOX bubble in there, rising up into the LOX tank and bursting, would have been extremely undesirable. $\endgroup$ – Organic Marble May 2 '17 at 20:45
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    $\begingroup$ Do we have any indication SpaceX uses this helium cooling method? $\endgroup$ – Hobbes May 3 '17 at 8:22
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    $\begingroup$ @Hobbes Quite late to the party but Helium bubbling was confirmed by Musk in 2016: "Launch aborted on low thrust alarm. Rising oxygen temps due to hold for boat and helium bubble triggered alarm." (via twitter) $\endgroup$ – Christoph Nov 15 '19 at 10:42

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