I see, within the holiwars around FH, which took and did not explode on the launcher table, and quite even sent something to the heliocentric orbit, one of the trump cards on one of the sides “and why with the claimed payload mass of 60+ tons FH pulonul only Tesla weighing a ton? ”
When trying to figure out the answer to this question, a couple of interesting details emerged.
We start with the load capacity of FH as a whole. On the SpaceX website, it is listed at 63.8 tons in low orbit (DOE), 26,700 kilograms per geotransport and 16,800 kilograms at the Homan flight to Mars.
However, on the slide from the presentation of the BRF Mask in the autumn of last year, we see 30 tons of DOE
Yeah, well, right, it’s “reuse”, that is, the return of steps to the landing platforms. The truth is not clear which of the numerous options for the return of central blocks to land or sea areas. The payload for DOEs has more than halved, but how much has the payload on the departure track decreased? The forum nasaspaceflight at one time made an assessment for an earlier version of FH
Here the first column is the presence or absence of a fuel overflow between the side blocks and the central one, as we know in the end, this was abandoned altogether. The second column is “RTLS” – the number of steps returning to the start, the third “ASDS” – sitting on offshore platforms. Those. our case is 4 line. Output to Mars PN is 4 tons. The correction factor for the characteristics FH grown from the moment of drawing up the plate can be taken from the last line – 11 tons are output to mars, and on the site 16.8, i.е. For the current version of FH, the numbers of the table should be multiplied by 1.5 times. It turns out that 6 tons was supposed to go to Mars, and about 1-1.5 went? Not enough.
In fact, this estimate is slightly optimistic. The separation of the central block and the upper stage was assumed here at a speed of ~ 3 km / s, cm corresponding to the simulation.
And 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 point. The fact is that every person who has not been cleaned up by the last step when detaching hundreds of meters per second is quite painfully bent on reducing the payload, it falls exponentially with the growth of the speed necessary for completing. It would be optimal to detach the central unit much later, at a speed of, say, 4 km / s, in general the most optimal missile has roughly the same distribution over the speed dialed by each speed.
However, putting the first stages at high speed is very difficult, if not impossible. When the Falcon Heavy was launched, the bet was obviously made not on the output of the maximum payload, but on the coverage of as many scenarios as possible related to the missile’s flight – there was a return of 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 did not do the maximum, in order to see how it behaves when returning. Which, of course, affected the load.
But in order to find out exactly what PN could bring FH in this flight, you need to know the final speed to which Tesla Roadster was dispersed. And it can be recognized
Here is a first version of the trajectory of the PN (which is then refined, but the required parameters are the same), and there is a wonderful tsiferka C3 Earth (km 2 / c 2 ) = 12.0 – is the excess of the speed of the second space velocity (ie, 11 , 2 km / s). The speed V pn that the stage receives is related to the second cosmic V 2 and C 3 by the following equation:
C 3 = V pn 2 – V 2 2
This year there should be a close convergence of the Earth and Mars, so to reach (at the start in April-May) it is possible with C3 of 7.7-7.9 km 2 / c 2, and PN Heavy on the departures to Mars was considered either for this figure, or for the average annual C3 = 9.5. A bit about this is told here in this video:
Total we get the following values:
1. The speed dialed by Tesla Roadster in the launch on February 6: V pn = sqrt (11,19 ^ 2 + C3) = 11,71 km / s The speed dialed by Falcon Heavy for the “standard” Mars should be 11.61 km / s, and for this year 11.53 km / s. To the ballistic speed it is necessary to add gravitational losses – to the eye about 200 meters per second, the obtained value is called the characteristic velocity or delta V in western terms.
Now let’s try to compare the difference in the characteristic speed, dialed by the “standard shipment” of cargo to Mars and the launch on February 6:
11.61 + 0.2-3 = 8.81 km / s <-> the step should output ~ 6 tons
11.71 + 0.2 – 2.64 = 9.27 km / s <-> the step was taken by a red typewriter with a dummy.
Here, more or less rigorous calculations are completed and shaky assumptions begin. For example, the speed of separation of the upper stage of FH under the most severe conditions of saving the central unit can be 3 km / s – this figure occurs in discussions at the nasaspaceflight forum, but the truth, of course, is known only in SpaceX. Another undetermined quantity will be the masses of the filled and empty upper stage FH, which have never been published either. Finally, the mass of the launched roadster is also unknown. Nevertheless, a plus or minus kilometer estimate can be made, and estimate which share from the full PN was launched on February 6.
So, for starters, the Tsiolkovsky formula
V = I * ln (Mn / Mk), where V is the characteristic velocity, I is the specific impulse (3355 m / s for Merlin 1D vac), Mn / Mk is the ratio of the initial mass to the final one.
For V (8.81) Mn / Mk = 13.81
For V (9.27) Mn / Mk = 15.85
For all the same nasaspaceflight I found such figures for the upper stage FH – the weight of the charged 95,000 kg, the weight of the spent stage is 3900 kg the numbers do not match the simulation above). For Mn / Mk = 15.85, this gives the maximum mass of the MON at the launch on February 6 at 2,093 kg, which is only slightly larger than the roadster without batteries, but with an adapter, transmitter and dummy. Quite an expected stock for unknown dynamic loads, etc. But the PN with the “standard” sending to Mars will be only ~ 3000 kg instead of 6000 of the estimates above and this is expected. To recruit 8-9 kilometers per second with one kerosene step, even such a masterpiece from the point of view of the ratio of empty weight to weight filled, means to destroy almost the entire weight of the MON in favor of the weight of the step.
It turns out that in such a carrier configuration, the FH’s ability to send something to Mars is rather modest – less than that of Delta IV Heavy, Atlas V 541, Proton-M / Breeze-M and Arian V, and it is determined by the extremely sub-optimal distribution of the characteristic velocity on the steps. In the run 06.02.2018 FH worked close to the limit of possibilities, if we start from the realistic figures of dry and full weight of the upper stage, which walk in the Internet.
The main conclusion is that Falcon Heavy has not yet shown its capabilities, and the next launches will collect data that will allow us to estimate the real weights of the steps and more accurately assess the possibilities of a reusable and one-time option.