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I was just reading about the Scout, an all-solid rocket, and thought of Rocket Lab's Electron, which seems to have a similar mission of deploying small payloads to orbit. I understand the engines to be the most expensive part of a rocket (especially the turbopumps), and the Electron has nine(!) Rutherford engines on the first stage. Since Rocket Lab has no plans for reusable launchers, it just seemed that a solid first-stage, with no plumbing or pumps, would be a lot cheaper.

But I can see the value of liquids for upper stages because they are more controllable, making it easier to fine-tune an orbit. And the Electron second stage only has one engine, anyway.

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    $\begingroup$ I think this answer might offer some insight. I'd say that the points made about ISP and fuel fraction sums up into low payload fraction - a big rocket for a small payload. $\endgroup$ – Anthony X May 11 at 13:14
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    $\begingroup$ For a rocket that uses this concept you can look at the Vega rocket build by Avio and launched by ArianeSpace. Their soon-to-exist Vega-C will use the Ariane 6 boosters as first stage. $\endgroup$ – GittingGud May 13 at 10:38
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    $\begingroup$ @GittingGud talking about other rockets that use SRB's as first stage, Ares I must be mentioned. $\endgroup$ – Everyday Astronaut May 16 at 6:02
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The Rutherford engines produce much less vibration than SRB's. This is actually a major selling point for Rocket Lab, as pointed out by Peter Beck here after 31:03:

In essence, the low vibration spectrum enables the customer to put more useful payload onboard the Electron, because they need less/lighter mechanical structure. This is especially important for small satellites where there are very tight limitations to the mass of the satellite.

This is also mentioned in the answer linked in @AnthonyX's comment. Since this here is a distinct question, and Rocket Lab is setting new standards in this field, I think it's worth being posted as a separate answer.

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    $\begingroup$ Nice answer! Related: Why does PSLV use four stages to get to LEO, and why do they alternate solid, liquid, solid, liquid? $\endgroup$ – uhoh May 11 at 22:54
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    $\begingroup$ That's a new direction for me. I had thought a satellite was developed and then a launcher was selected. Rather, choose a launcher, then design your satellite to that standard. $\endgroup$ – Greg May 11 at 23:15
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    $\begingroup$ @Greg, more you design your sat, then find out how much supporting equipment needs to be added to mate it to a given launcher to get it safely to orbit. Not Electron specific but this also comes up if your sat needs power or coms from the launch vehicle, not all launchers provide the same services to the payload. $\endgroup$ – GremlinWranger May 11 at 23:26
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Solid Rocket Motors are inexpensive to manufacture if you have the knowledge and experience already. For space use they are commonly made by weapons makers, sharing many production assets with weapon manufacturing.

Minus the military infrastructure, they are not as easy or economical. A private company like Rocket Lab would likely have to buy solid rocket motors from an outside source, leaving them subject to supply delays, someone else's pricing, etc. Alternatively they could develop the capability in-house, with a large increase in workplace hazards, large R&D expenses.

Instead they pursued a newer cost reduction scheme than the classic 'bundle defense and aerospace buying in Solid Rocket Motors' - They designed a simplified liquid fuel engine that bypassed the typically complex task of designing and building turbo-pump hardware that can withstand exhaust gasses and pressures. They use 3d printing, high volume assembly line production, and battery powered pumps to build inexpensive and efficient liquid fueled engines.

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In addition to Answers by Everyday Astronaut and Saiboogu a potential advantage is lower overall program cost due to safety. The current Electron is very safe in fully assembled form until fueled so can be tested, observed by media/public and moved around with minimal difficulty.

Then once tested and ready to go the fuel can be loaded remotely and launched. Or it fails to launch fuel can be remotely drained and then investigators can approach a reasonably inert rocket. For solids as soon as the compound is mixed you have a fire/explosion hazard that requires a complex chain of safety systems to manage, and can never be completely safe (for example against sabotage) only 'as low as possible'. While liquid oxygen production has had it's own problems there is this incident.

If you have a local solid fuel production facility all these problems become smaller since you can fill on demand, if you want to launch out of New Zealand or the UK your ground safety process start to get expensive with long lead times and the need to store spares.

It may also be easier to write ecological impact statements for a ground/fire explosion for the current fuels than for a solid rocket making approval easier to sell politically.

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