An interesting observation on the launch of Falcon Heavy

I see, within the framework of holivars around FH, which took and did not explode on the launch pad, and quite even sent something to a heliocentric orbit, one of the trump cards from one of the parties "and why, with a declared payload mass of 60+ tons FH pulnul just one Tesla weighing a ton? "

When trying to figure out the answer to this question, a couple of interesting details emerged.

Let's start with the load capacity of the FH as a whole. On the SpaceX site, it is indicated at 63.8 tons to a low orbit (LEO), 26,700 kilograms per geotransfer and 16.800 kilograms per Gomanian departure to Mars.
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However, in the slide from the BRF Mask presentation last fall, we see 30 tons at the NOU



Yeah, well, right, it's “reuse”, I mean the return of the steps to the landing platforms. True, it is not clear which of the many options for returning the central units to land or offshore sites. The LEO payload has more than doubled, but how much has the payload on the departure trajectory decreased? At the nasaspaceflight forum, in due time they made an assessment for an earlier version of FH

Here, the first column is the presence or absence of fuel overflow between the side blocks and the central one, as we know as a result, this was completely abandoned. The second column “RTLS” is the number of steps returning to the start, the third “ASDS” - landing on offshore platforms. Those. our case is line 4. Mon brought to Mars - 4 tons. The correction factor for the FH characteristics that have grown since the tablet was compiled can be taken from the last line — 11 tons are output to Mars, and 16.8 on the site, i.e. for the current version of FH, the numbers in the table should be multiplied by 1.5 times. It turns out that 6 tons should have gone to Mars, and about 1-1.5 went? Not much.

In fact, this estimate is slightly optimistic. The separation of the central block and the upper stage here was assumed to be at a speed of ~ 3 km / s, see the corresponding simulation.



But in reality it happened at a speed of ~ 2.64 km / s.



Most likely, in this difference between simulation and reality there is a rather interesting moment. The fact is that every time when hundreds of meters per second are not picked up by the last step, it hurts about the reduction of the payload, it decreases exponentially with the speed necessary for the extraction. It would be optimal to separate the central unit much later, at a speed of, say, 4 km / s, in general, the most optimal rocket has approximately the same distribution in the step gained by each speed.

However, it is very difficult, if not impossible, to plant the first steps at high speed. When launching the Falcon Heavy, the stake was clearly not made on outputting the maximum PN, but on covering as large as possible scenarios related to the rocket flight with the test - here and returning the side blocks to the ground and landing on the offshore platform (unsuccessful). Continuing this logic, we can assume that most likely the speed of separation of the central unit was not maximized, in order to see how it behaves when it returns. Which, of course, affected the load.

But in order to find out exactly which PN could take FH on this flight, you need to know the final speed to which the Tesla roadster was accelerated. And you can recognize her



Here is shown the first version of the PN trajectory (which was later clarified, but the parameters we needed remained the same), and there is a remarkable tsiferka C3 Earth (km ² / c ² ) = 12.0 - this is an excess of speed over the second cosmic velocity (i.e. 11 , 2 km / s). The speed V _pn , which the stage is gaining, is related to the second cosmic V ₂ and C3 by the following equation:

C3 = V _pn ² - V ₂ ²

This year there should have been a close approximation of the Earth and Mars, so to reach (at the start in April-May) it is possible with C3 7.7-7.9 km ² / c ² , and the MO Heavy for departure to Mars was considered either for this figure , or for average multi-year C3 = 9.5. A little about this is told here in this video:



Total we get the following values:

1. The speed gained by Tesla by the Roadster at the start of February 6: V _pn = sqrt (11,19 ^ 2 + C3) = 11.71 km / s. The speed gained by Falcon Heavy for the “standard” PN transfer to Mars should be 11.61 km / s, and for this year 11.53 km / s. It is necessary to add gravitational losses to the ballistic speed - by eye about 200 meters per second, the resulting value is called the characteristic speed or delta V in western terms.

Now we will try to compare the difference in the characteristic speed, recruited by the “standard dispatch” of the load to Mars and launch on February 6:

11.61 + 0.2 - 3 = 8.81 km / s <-> the stage should output ~ 6 tons
11.71 + 0.2 - 2.64 = 9.27 km / s <-> the stage brought out a red typewriter with a dummy.

Here, more or less strict calculations end and shaky assumptions begin. For example, the speed of separation of the upper FH stage under the most severe conditions of salvation of the central unit may be 3 km / s - this figure is found in discussions on the nasaspaceflight forum, but the truth, of course, is known only in SpaceX. Another uncertain quantity will be the masses of the filled and empty upper stage FH, which have also never been published. Finally, the mass of the launched roadster is also unknown. Nevertheless, it is possible to make a plus or minus kilometer estimate, and estimate what share of the full MON was launched on February 6.

So, to begin with, Tsiolkovsky’s formula

V = I * ln (Mn / Mk), where V is the characteristic speed, I is the specific impulse (3355 m / s for Merlin 1D vac), Mn / Mk is the ratio of the initial mass to the final mass.

For V (8.81) Mn / Mk = 13.81
For V (9.27) Mn / Mk = 15.85

At the same nasaspaceflight, I found such figures for the upper FH stage - the mass filled with 95,000 kg, the mass of the spent stage is 3900 kg (the figures do not coincide with the simulation above). For Mn / Mk = 15.85, this gives the maximum PN mass at launch on February 6 of 2093 kg, which is only slightly larger than a roadster without batteries, but with an adapter, a transmitter and a dummy. Quite expected margin on unknown yet dynamic loads, etc. But the PN with the “standard” dispatch to Mars will be only ~ 3000 kg instead of 6000 from the ratings above and this is expected. To gain 8-9 kilometers per second with a single kerosene stage, even such a masterpiece in terms of the ratio of empty weight to the weight filled, means to destroy almost the entire weight of the MON in favor of the weight of the stage.

It turns out that in such a configuration of the carrier, the FH capabilities for sending something to Mars are rather modest - less than that of Delta IV Heavy, Atlas V 541, Proton-M / Briz-M and Arian V and this is determined by the extremely non-optimal distribution of the characteristic speed on the steps. At the launch of 02/06/2018 FH worked close to the limit of possibilities, if we start from the realistic figures of the dry and full weight of the upper stage, which are walking on the Internet.



The main conclusion is that Falcon Heavy has not yet shown its capabilities, and the following launches will allow us to collect data that will allow us to estimate the real weights of the steps and more accurately assess the possibilities of the reusable and one-time option.