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Suppose we wanted to do a very small experiment in (or very close to) the Cosmic Microwave Background Radiation rest frame, which we are travelling at 368 ± 2 km/s relative to. Without considering the complexity of measuring and transmitting the results back home, how would we get a 1 gram payload to such a high delta-v in the first place? What would it cost?

Is there a way to naively extrapolate costs based on required delta-v?

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    $\begingroup$ 368km/s is far beyond our current capabilities, even using ion engines or other in-development high efficiency engines. So assigning a monetary value to it will be fairly meaningless, but maybe still a fun exercise! $\endgroup$ – Jack Aug 25 '18 at 7:30
  • $\begingroup$ It can't be done by rocket engines currently. You can exercize yourself using Rocket equation en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation. Note that main problem is we can't reach high enouth exhaus velocity, even with ion engines. I guess maybe some magnetic levitatgion loop in vacuum theoretically could... $\endgroup$ – Heopps Aug 25 '18 at 7:52
  • $\begingroup$ @Jack I think you are overstating the hopelessness/meaninglessness situation. While it's not something you can order out of a catalog yet, there should be enough work on practical realizations of solar sails to make a guesstimate. See How will The Planetary Society's LightSail 2 Spacecraft's retroreflectors be used? It is also possible to estimate how thin the sail would be to reach this speed (about 0.1% c) based on this answer. $\endgroup$ – uhoh Aug 25 '18 at 8:27
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    $\begingroup$ @Jack This speed is way way lower than the speeds considered by the Breakthrough Starshot project, so if done with a laser instead of sunlight, it wouldn't need anything near as big as Breakthrough Starshot's 100 Gigawatt laser array - what's the current thinking how this might work? $\endgroup$ – uhoh Aug 25 '18 at 8:29
  • $\begingroup$ @uhoh Agreed, but these technologies are still in development so extrapolating beyond the very small data set of tested prototypes and giving a monetary value would be speculative, I believe. I'm writing an answer for conventional engines, you could write one covering the techs you mentioned? $\endgroup$ – Jack Aug 25 '18 at 8:44
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With current technologies, this is unfortunately well outside our reach. However, there is promise on the horizon!

Chemical Engines

The Tsiolkovsky equation is always your friend when calculating Δv for conventional engines (or your enemy, depending how you look at it!):

$$ \Delta v = I_{sp} \times g \times \ln \frac {Mass_{full}} {Mass_{dry}} $$

Rearranging to solve for fuel ratio gives us:

$$ Ratio = \frac {Mass_{full}} {Mass_{dry}} = e^{\frac{\Delta v}{I_{sp} \times g}} $$

It's that exponential that causes us problems. Even if we use one of the most efficient chemical engines in history, the Space Shuttle Main Engine ($I_{sp}$ ~ 452s), ignore its mass and ignore the mass of all the tanks/plumbing/other structure, we get a lower-bound of $Mass_{full}\approx10^{33}$kg or 1000 times the mass of the Sun. When we include all the required structure to hold all this fuel, this gets even worse!

We could cut this number significantly by making use of staging, but it's clearly not going to give us anything possible, let alone affordable. So we have to go for higher efficiency.

High-efficiency engines

If we use one of the highest efficiency engines flown, Dawn's ion thruster ($I_{sp}$ ~ 3100s), and include the mass of the engine and tanks (8.2kg engine, tanks based on square-cube from 450kg fuel : 19kg tanks), we get $Mass_{full}\approx 5\times10^{14}$kg - still totally unfeasible.

But we can do better.

ESA's in-development Dual-stage Gridded Ion Thruster (DS4G) has been calculated as achieving an $I_{sp}$ of around 20000s.

Swapping Dawn's ion propulsion ion engine for one of the same mass with an $I_{sp}$ of 20000s will get us an enormous 82km/s! If we add more fuel and scale the tank's mass accordingly, we can achieve our 368km/s with a total craft mass of ~6000kg - totally achievable!

Dawn cost around \$450m, so I'd speculate a very rough conservative cost of \$1b for building and launching our hypothetical craft. Economy of scale saves us money on the larger mass and the launch costs won't be significantly more. This obviously ignores any costs from developing the dual-stage technology which would be very difficult to estimate.

Other technologies

We can see that whatever we try, the rocket equation is always going to bite us at some point, so why don't we try something that doesn't require propellant?

Breakthrough Starshot is a proof-of-concept technology that can supposedly achieve speed far in excess of our 368km/s - on the order of 0.1c! It uses Gigawatt (read: peak power draw comparable to large countries) ground-based lasers to propel tiny crafts with extremely high acceleration.

This kind of propulsion would be ideal for your proposal - the craft would reach the required speed in a very short time, minimising corrections needed for gravitational influences and negating the need for large transmission systems.

The kind of infrastructure infrastructure would clearly be incredibly expensive - probably on the magnitude of the infrastructure budget of whole countries - \$100b - \$1t.

However, Breakthrough Starshot is relying on the costs of components dropping significantly and efficiencies increasing as the technologies progress. Some estimates give a single mission cost of $5-10b in 2036, with speculative drops in cost. Again, this doesn't account for the cost of research and development.

Note - I've tried to make some speculation and estimates on the costs involved, but they should all be taken with a pinch of salt. Also, since the 1g payload is unspecified, I'm assuming it can be modified to suit the requirements of the craft

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    $\begingroup$ Assuming everything is in orbit already (having used staging and an extreme amount of launches to assemble in orbit), using the STS main engine, and a 10% tank weight (that's nominal, right?), how much fuel to get to Δv 368 km/sec? And how many Saturn Vs (or today's equivalent; w\e is better) would it take to put all that fuel into orbit? $\endgroup$ – Mazura Aug 25 '18 at 15:56
  • $\begingroup$ @Mazura significantly more than $10^{33}$kg which is the lower bound for the fuel mass assuming a massless engine and massless tank! If you give them both masses, the fuel requirements get even larger. If we just include the mass of the engine (3500kg) and ignore the tank, the required fuel already jumps to $10^{39}$kg! See here $\endgroup$ – Jack Aug 25 '18 at 17:21
  • $\begingroup$ Only 368 km/s? I thought that was just in range of Ole Boom Boom. They got as far as building and launching a test vehicle. $\endgroup$ – Joshua Aug 26 '18 at 4:11
  • $\begingroup$ I suspect a powered gravitational slingshot maneuver around the Sun could do it. The top speed of the Parker Solar Probe will be 200 km/s, probably it wouldn't be very hard to tune it to 368 km/s and into the correct orbital plane. $\endgroup$ – peterh says reinstate Monica Aug 26 '18 at 18:42
  • $\begingroup$ This simple form of the Tsiolkovsky equation can cause trouble that you allude to: "When we include all the required structure to hold all this fuel, this gets even worse!" I see you've wisely deleted reference to the square-cube law, which doesn't hold for tanks of a specified internal pressure. My answer to the question, "Is there a ∆V limit for a single-stage rocket with propellant tank mass proportional to propellant mass?" treats this. (There is a limit that cannot be exceeded!) $\endgroup$ – Tom Spilker Aug 27 '18 at 19:54
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Jack did a great job describing how to do it using propulsive engines. I have a different answer:

We already (almost) planned to do it (inadvertently).

The original plan for the recently launched Parker Solar Probe was to do a gravity assist at Jupiter for a subsequent fly-by of the Sun at a relative speed of more than 300 km/s. So, to get to a speed of 370 km/s instead would require a distance to the Sun of 3.5 instead of 4 solar radii - totally doable if we don't have to deal with sensitive instruments that need to be shielded from the intense radiation and heat.

Now we just have to make sure that the velocity vector of the probe is properly aligned with the CMB, but this is possible: The inclination can be varied by aiming at different edges of the Sun while the direction within the ecliptic just needs a proper timing with respect to the position of Jupiter.

Unfortunately, this maneuver provides the "being at rest w.r.t. the CMB" requirement for the "small experiment" only for one instant in time, and not for an extended period. If you need that, we're back at Jack's answer.

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    $\begingroup$ @uhoh: "far beyond our current capabilities" is definitely right, if you want to have the payload at rest with the CMB for an extended period of time. The solar flyby provides it for an instant only, which is not too useful if you want to do an experiment. $\endgroup$ – asdfex Aug 25 '18 at 15:22
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    $\begingroup$ On the other hand, 3.5 solar radii is maybe not quite the sort of dark and cold location that is best suited to CMB measurements. $\endgroup$ – E.P. Aug 26 '18 at 17:30
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    $\begingroup$ @E.P. is right. You're trying to measure, very accurately, the radiative flux from the CMB at 2.725 K. Things at 2.725 K don't radiate much (power goes at T^4), so the signal is exquisitely small. This is why the instruments making those measurements take a long time to do it: they have to integrate for a long time to build up enough SNR. A location where the free-body temperature of a high-albedo object is over 2,000 K is no place for those measurements, and you don't have a lot of time—mathematically, zero window duration!— to integrate. $\endgroup$ – Tom Spilker Aug 27 '18 at 5:40
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    $\begingroup$ @TomSpilker Nobody said you actually want to measure CMB there - maybe it's just a philosophical experiment to feel at rest with the universe ;-) $\endgroup$ – asdfex Aug 27 '18 at 11:16
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    $\begingroup$ @asdfex Point taken. But I'll bet you'd have a lot more trouble getting funding for the "philosophical experiment" than for CMB measurements! ;-) $\endgroup$ – Tom Spilker Aug 27 '18 at 19:47

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