This is from the latest Toughsf post: http://toughsf.blogspot.com/2019/11/hypervelocity-macron-accelerators.html
Hypervelocity Macron Accelerators
We look at the various ways of accelerating micro-scale projectiles up to hypervelocity (10-10,000 km/s) and their use in space.
Going small to go fast
Macrons or macroscopic particles are tiny projectiles that sit on the border between the complex structures we see under a microscope and the far simpler molecules where we can count individual atoms.
A typical macron is a micrometre in diameter and has a very simple structure. Due to the small size, it exhibits an interesting feature: a very high surface area to mass ratio. A useful number of electrical charges can be placed on a macron’s exterior compared to how much they mass. This feature can be exploited by an electrostatic accelerator.
Tiny particles are too small to survive the heating and friction in a railgun, and cannot support large magnetic fields in a coilgun. However, an electrostatic accelerator can bring particles up to high velocities by using a voltage gradient between an anode and a cathode. Charged particles feel a force when placed in between electrodes, proportional to the voltage gradient multiplied by the particle’s charge. When we divide that force by the mass of the particle, we get force divided by mass, which is an acceleration. A macron, with its high charge to mass ratio, will experience a strong acceleration even under small voltages.
The velocity gained by a non-relativistic charged particle is easy to calculate:
Particle Velocity = (2 * Voltage * Charge/Mass)^0.5
The velocity will be in meters per second.
Voltage is in volts.
Charge is in coulombs. Mass is in kg.
Charge to Mass ratio, the critical feature of macrons, is in C/kg.
Electrostatic acceleration is regularly used today to push small things to great speeds. For example, electric rocket engines such as colloid thrusters shoot out tiny liquid droplets at multiple kilometres per second, which is somewhat similar to how we want a macron accelerator to operate. One design accelerates them to 43km/s. We can also find electrostatic accelerators in the medical field. In fact, the majority of them today are used to generate X-rays for therapy.
The most powerful electrostatic accelerators are for nuclear research purposes. They operate at several megavolts and are used to accelerate electrons and ions.
Van de Graaff accelerators have been used to study the impacts of interplanetary dust grains. The original accelerator facility built by Friichtenicht in 1962 was able to accelerate 0.1 μm iron spheres to 14 km/s using a 2 MV potential. We have also designed Cockroft-Walton, Marx and Pelletron accelerators, each different in their way of creating and holding a large voltage potential.
How much voltage can be obtained in an accelerator?
A voltage ‘V’ between two surfaces separated by a distance ‘m’ will create a voltage gradient of V/m. 1,000 Volts across a 1 centimetre gap will give a gradient of 1,000/0.01 or 100,000 V/m. The gradient creates a force that accelerates charged particles, but can also give electrons within the two surfaces enough energy to jump across the gap. The electrons that escape gain energy and slam into the opposite electrode, damaging it and reducing its ability to maintain a voltage gradient.
Weak voltage gradients are enough to get electrons to jump across a gap filled with a conductor, like salty water. Stronger gradients are needed to cross insulated gaps, like pure vacuum. Charge will accumulate at the tip of any imperfections or contaminants on the surface of an electrode, creating stronger voltage gradients locally and reducing the overall voltage that is possible. Too high a gradient, and enough electrons jump across to create an electric arc. The arcs from too much voltage gradient are lost energy that does not go towards accelerating charged particles. In fact, that energy becomes heat that can burn out anodes and cathodes. A device that operates at the megawatt or gigawatt level will certainly not want electric arcs dissipating that power as heat internally!
Single stage accelerator today manage 10 to 15 MV in total. Getting more than that becomes exceedingly troublesome, as voltage multiplying circuits become larger and larger.
Tandem (or two stage) electrostatic accelerators double their maximum voltage by switching the charge on the particle being accelerated halfway down their length.
At the upper end, we see Pelletrons with 30 MV. However, the highest voltages are only possible because the electrode gap is filled with a pressurized insulating gas. We cannot use this option inside our macron accelerator as interactions between the charged particle and the gas could create enough friction heating to destroy it. We are therefore forced to rely on simple vacuum. Highly charged macrons cannot be quickly switched from a negative to a positive charge either; the charges that need to move quickly result in destructive currents. Tandem accelerators are not an option either.
A ‘Super Marx Generator’ was proposed for achieving voltages on the order of 1000 MV; it is 1500 meters long and therefore averages 0.6 MV/m. That design stored a gigajoule of energy. Our macrons do not need that much energy but the length requirements are similar; 100 MV would require 167 meters of capacitors in series.
Another option is a pulsed staged accelerator. An electrostatic accelerator can be broken down into a series of stages. Each stage consists of a pair of charged plates with a gap between them. The voltage gradient between each pair of plates, assuming excellent vacuum, no contamination and perfectly smooth surfaces, can be on the order of 1 to 10 MV/m, which is far better than the Super Marx generator design. More realistically, 3 MV/m is achievable.
If an electrical current is supplied to the pairs of plates for just the short period where a macron crosses through them, and then it switched off as the macron exits, we end up accelerator driven by pulses of electricity with multiple stages. This design was suggested and demonstrated (for 5 stages) here. The increased voltage gradient means that 100 MV only needs between 10 and 100 meters of length.
The downside is that switching the plates on and off is not a perfect process. The switches, likely to be solid state transistors, convert some of the electrical energy into heat and can become a major source of inefficiency.
We can also consider a circular accelerator.
Instead of thousands of stages, a single stage is reused multiple times. The macrons will be bent 180 degrees twice by U-shaped magnets to form a looping trajectory. They gain velocity with each revolution, so there is no ‘maximum voltage’. However, we cannot increase the velocity past the point where the magnets cannot bend the macron’s trajectory.
The maximum velocity achievable can be calculated with this equation:
Maximum Velocity = Bend Radius * Magnet Strength * Charge/Mass ratio
Maximum Velocity is in m/s.
Bend Radius is in meters.
Magnet strength is in Tesla.
Charge/Mass ratio is in C/kg.
All of these factors affect velocity linearly. You will notice that the voltage gradient does not come into play at all; it simply takes more revolutions to reach the maximum velocity if the voltage is weaker.
Large circles have the lowest magnetic strength requirements to reach a certain velocity. However, spacecraft might have certain constraints on their cross-section and size that prevents them from mounting circular accelerators above a certain radius, so a linear accelerator might be preferred for reaching high velocities. It should be noted that the velocity achieved in a linear accelerator is proportional to the square root of the charge to mass ratio, but it is directly proportional in a circular accelerator. This means that as C/kg values increase, the circular accelerator becomes more attractive.
In spacecraft with both length and radius restrictions, the circular and linear accelerators can work together to maximize the velocity possible.
After exiting an accelerator, macrons can be neutralized by passage through a thin plasma, and at the highest velocities, by a charged particle beam of the opposite charge. The tiny, rapidly cooling particle will become nearly impossible to detect or deflect until it hits a target.
Charge to Mass ratio
To reduce the weight of the accelerator, a lower voltage requirement is needed. To do this, charge to mass ratio must be maximized.
For a spherical macron, surface area to volume ratio increases at the same rate as radius decreases. A sphere with a radius 10 times smaller has a 10 times better surface area to volume ratio. This could mean a 10 times better charge to mass ratio.
A huge amount of charge can be added by various methods. How much charge can a sphere hold?
The total charge is given by:
Surface charge = 1.11*10^-10* Voltage Gradient * Radius^2
Surface charge is in Coulombs (C)
The voltage gradient within the projectile is in V/m
Radius is in m
The 1.11*10^-10 coefficient is (4*pi*Permittivity of Vacuum)
The charge divided by mass, or charge to mass ratio, for a sphere is:
Charge to mass ratio = 2.655 * 10^-11 * Vg/(Radius * Density)
Charge to mass ratio is in C/kg.
Vg is the voltage gradient within the projectile in V/m
Radius is in meters.
Density is in kg/m^3.
These equations show that want to maximize the charge to mass ratio, the radius has to be very small and the voltage gradient as high as possible. The maximum voltage gradient for a negatively charged particle is about 100 MV/m. For a positively charged particle, this increases to 1,000 MV/m. Other sources mention voltage gradients as high as 50,000 MV/m as being possible, but that is likely to be a theoretical limit. If the particle is charged too much, it will start releasing electrons through field emission and dissipating the excess potential charge. For tiny projectiles, this causes enough heat to destroy them.
We can suppose that any macron that we need to accelerate to very high velocities will be pushed to this limit.
Let’s take the example of a positively charged 1 mm wide iron sphere. Radius is half of the diameter, so 0.5 mm or 5*10^-4 meters. The density is 8600 kg/m^3. The maximum charge to mass ratio will be 0.006 C/kg.
Now let’s work out the C/kg for a positively charged 1 micrometre lithium sphere. Radius is 0.5 micrometres. Density is 534kg/m^3. The maximum charge to mass ratio for this macron is 99 C/kg.
The lithium macron is clearly superior to the iron particle, because it is much smaller and composed of a less dense material.
Another maximum is the strength of the macron’s materials. The voltage gradient creates a force that tensile strength must overcome. The equation for a macron stressed to the limits of its tensile strength is:
Strength-limited C/kg = (1.77 * 10^-11 * T)^0.5/(R * Density)
The charge to mass ratio is in C/kg.
T is the tensile strength in Pascals.
R is the radius in meters.
Density is in kg/m^3.
Using the previous examples, a 1mm iron sphere with a strength of 250 MPa could survive a charge to mass ratio of 0.015 C/kg. This is a quarter of the previous limit.
Lithium is rather weak with 15 MPa of tensile strength. A micrometre wide particle of lithium would only survive a charge to mass ratio of 0.99 C/kg, so it is strength limited to a hundredth of the previous value.
To achieve better C/kg, we need stronger materials. The tiny dimensions of macrons bring forward another advantage to help meet this requirement. At a very small scale, we can expect materials to be formed without any defects. This unlocks their full strength potential.
A good example of this small-scale advantage is iron.
Bulk iron has a strength of 250 MPa. However, a micrometre-long monocrystalline whisker of iron displays strengths of 14,000 MPa. The charge to mass ratio allowed by micro-scale iron’s strength is 0.12 C/kg.
This difference in strength between bulk and micro-materials can be demonstrated for graphite, aluminium, silicon and many others.
A silicon nitride whisker has a strength of 13,800 MPa and a density of 3200 kg/m^3. It can have a micro-scale charge to mass ratio of 309 C/kg.
The current champions of strength to weight ratio are carbon fibres. The Toray T1100G fibre is the strongest material commercially available for its weight, at 7,000 MPa for 1,790 kg/m^3. A micrometre-sized sphere of these fibres can support charge to mass ratios up to 393 C/kg.
At the microscopic scale, those same carbon fibres gain the incredible properties of carbon nanotubes. They have shown strength to weight ratios more than ten times better than Toray T1100G fibres (about 63,000 MPa for 1340kg/m^3), which means charge to mass ratios of at least 1,576 C/kg at the same scale.
How do we actually use the full potential of these small-scale materials if by making them stronger, we run again into the field emission limit on charge to mass ratio?
Shaping the macron
The solution to more C/kg is to move past simple spheres.
The spheres can be made hollow. This retains the surface area of a sphere but decreases the mass. We can call W the ratio of wall thickness to radius. W=0.5 means that the walls are half as thick as the sphere’s radius. W=0.01 means that the walls are a hundred times thinner than the sphere’s radius.
The wall thickness ratio can by multiplied against the density in the previous equations to give the field-emission-limited charge to mass ratio of a hollow shell:
Hollow C/kg = 0.02655 / (Radius * W * Density)
The charge to mass ratio is in C/kg.
R is the radius in meters.
W is the wall thickness ratio. Density is in kg/m^3.
This limit improves by a factor 1/W as the wall to thickness ratio decreases. At W:0.1, the field-emission-limited C/kg is increased by a factor 10. At W:0.01, it is a hundred-fold better.
However, a hollow shell has W times less thickness to resist forces, and also has W times less mass to support. The strength-limited charge to mass ratio becomes:
Hollow C/kg = (1.77 * 10^-11 * T * W)^0.5/(Radius * Density * W)
The charge to mass ratio is in C/kg.
T is the tensile strength in Pascals. W is the wall thickness ratio.
R is the radius in meters. Density is in kg/m^3.
Notice how this limit improves by a factor 1/W^0.5 as the wall to thickness ratio decreases. The benefit from W:0.1 is only 3.3x, and from W:0.01 is 10x better than a full-thickness sphere.
These equations for hollow spheres imply that as the walls get thinner, the strength of the projectiles becomes more important.
There is also the option to shape the macron into a cylinder.
Cylinders can be elongated to large length to width ratios, like in this paper. This gives them a better surface area to volume ratio than spheres.
The ratio between the lateral surface area of a tube of elongation G and a sphere of equal volume is:
Surface area ratio for cylinder vs sphere = 0.605*G^0.333
G is cylinder length divided by cylinder radius
A tube that is 1000 times longer than it is wide (G:2000), for example 1 um wide and 1 mm long, would have a surface area that is 7.6 times greater than a sphere of equal volume. Carbon nanotubes that are a few nanometres wide and up to several centimetres long would have G:10,000,000, so they are a 131 times better shape than a sphere.
Cylinders, of course, can be hollowed out to become tubes. The benefits of elongation and wall thickness ratio are multiplied in this case.
Ion beam for macron acceleration
An electrostatic accelerator can be used in an entirely different way to get a macron up to high velocities.
Instead of directly pushing and pulling on a macron using electric fields, it can act on it indirectly with a beam of electrons or protons. This has been called a ‘beam pushrod’ or a ‘beam blowpipe’.
The charged surface of a macron naturally repels objects of similar charge. If it has an internal voltage gradient of 1000 MV/m and a diameter of 1 millimetre, it can repel particles with an energy of up to 500 keV. A thousand times smaller particle can only deflect particles of up to 500 eV, but it will accelerate harder thanks to the square-cube law. A negatively charged macron would only produce internal voltage gradients of 100 MV/m, so it would be deflecting 50 keV beams at 1mm and 50 eV at 1 um.
Less energetic beams can be deflected further away from the particle, giving it a larger effective cross-section. For example, a macron that could deflect a maximum of 50 eV electrons would be able to deflect 25 eV with an effective cross-section twice its actual physical size. We will assume that electric or magnetic fields are used to focus the charged beam onto a spot equal to the size of the macron’s effective cross-section throughout the length of the accelerator, as has been proposed here. Low energy charged beams will tend to expand very rapidly once outside the focusing influence of these lenses, so acceleration past the last focusing element can be ignored.
The maximum acceleration that a macron can survive in these conditions is dependent on tensile strength:
Maximum acceleration = (0.75 * T) / (R * Density)
Maximum acceleration is in m/s^2.
T is tensile strength is in Pascals.
R is macron radius in meters.
Density is in kg/m^3.
The value is independent of the wall thickness ratio.
A millimetre-sized projectile made of a material such as aluminium 7075-T651 (570 MPa, 2800 kg/m^3) could be accelerated at up to 1.52*10^8 m/s^2.
Meanwhile, a micrometre-sized sphere of diamond (1600 MPa, 3510 kg/m^3) would accelerate at 3.42*10^11 m/s^2.
To accelerate a positively charged macron, a proton beam would be used. At 500 keV, protons have a velocity of 9,780 km/s. At 500 eV, this falls to 300 km/s.
A negatively charged macron can be pushed by an electron beam. Electrons with an energy of 50 keV travel at 123,000 km/s. At 50 eV, it is 4190 km/s.
These figures do not mean that the macron can only asymptotically approach the beam’s own velocity. They are the maximum relative velocity between the beam and the macron. If the macron is already travelling at 4190 km/s (50 eV electrons), then it can actually deflect 100 eV electrons (5929 km/s). A series of pulses from a particle accelerator, each tuned to have an energy that closely matches that of a macron, can bring that macron up to higher and higher velocities in steps. This is also good for transferring momentum to the macrons efficiently.
Pushing a macron with a charged beam has the advantage that almost any beam intensity can be used. Since the protons or electrons do not touch the macron and are instead deflected electrostatically, none of their energy is converted into heat. Also, the accelerating tube can be equipped with electrostatic or electromagnet lenses that can focus a charged beam and maintain high intensity throughout the duration of the acceleration. The beam energies are relatively low, so the focusing elements can be very lightweight and the acceleration tube extended without great mass penalties.
Other than the strength of the macrons, the limit on ‘pushrod’ acceleration is the beam’s charge density.
Protons or electrons do not like being bunched up behind a macron. They repel each other. If we can only put a certain number of charged particles behind a macron (the current density), we can only deliver so much energy, which limits acceleration.
For a pulse of non-relativistic protons and electrons bouncing off a macron, the maximum current density is given by the Child-Langmuir Law:
Maximum Current Density = (7.7*10^6*BE^1.5*R)/(Pulse Duration* BV)
Current density is in Amperes per square meter (A/m^2)
BE is beam energy in electronvolts (eV)
R is macron radius in meters
Pulse Duration is in seconds
BV is Beam Velocity in meters per second.
For a 500 keV beam of protons composed of 1 nanosecond pulses, pushing on a micrometre-sized particle, we have BE: 500,000 eV, radius 0.5*10^-6 meters, pulse duration 10^-9 seconds and BV is 9,782,000 m/s. The maximum current density becomes 1.39*10^12 A/m^2.
For a 50 eV beam of electrons composed of 1 microsecond pulses, pushing on a millimetre-sized particle, we have BE: 50 eV, radius 0.5*10^-3 meters, pulse duration 10^-6 seconds and BV is 4,193,200 m/s. Maximum current density becomes 3.24*10^6 A/m^2.
To maximize intensity, and therefore acceleration, we want the shortest pulses of the highest energy protons.
We can simplify the process for finding out the acceleration of a spherical macron by working with the power delivered by the pulses:
PA = ((0.375 * Current Density * BE)/(R * Density * PD * W))^0.5
PA is Pulse Acceleration in m/s^2.
Current density is in A/m^2.
BE is beam energy in eV.
R is macron radius in meters.
Density is in m^2
PD is Pulse duration in seconds.
W is the wall thickness ratio.
Following on from the previous examples:
A 500 keV beam pushing a micrometre-sized particle made of diamond (3510 kg/m^3) with nanosecond pulses of protons would provide an acceleration of 3.85 * 10^15 m/s^2. This is a value greater than the maximum the hollowed-out diamond macron could survive mechanically, as calculated above.
A 50 eV electron beam pushing on a sphere of aluminium a millimetre wide achieves an acceleration of 1.75*10^7 m/s^2. This is lower than the maximum the macron can handle.
The macron’s shape could also be improved for use in this type of accelerator. A flat disk catches a larger beam, and so more energy could be transferred with each pulse. A web of fibres, inspired by the designs for electric sails, could have exceedingly high beam capture areas for their mass.
Producing the beam that pushes the macrons is generally not a challenge. Low energy electron beam specifically can be very lightweight, efficient and small. If we base ourselves on the designs of inductive output tubes, 15 kW/kg at over 80% efficiency is to be expected from today’s technology. Proton beams are trickier to produce, but they will still be small and lightweight in absolute terms. Their energy will come from the same RF generators as mentioned previously. See the Particle Beams in Space post for more details.
Since only one particle can be accelerated in a ‘pushrod’ accelerator at a time, it would make sense to also have those generators feed a multitude of accelerator tubes in sequence. 10 generators, each capable of producing 1 GHz of nanosecond pulses, could feed 10 tubes with a continuous supply of pulses each; if each macron clears a tube in 0.1 milliseconds, then ten tubes would have a maximal firing rate of 100,000 projectiles per second.
This might seem like a lot, but each projectile is expected to carry very little energy. A nanogram at 1000 km/s is still only 0.5 joules. A hundred thousand of them per second is just 50 kW. Accelerators in the megawatt range would end up looking like a volley gun or a ‘Metal Storm’ launcher.
Here is a selection of macrons to represent the different approaches to maximizing their performance
1) A 1 mm diameter sphere of diamond with 1,600 MPa strength and 3510 kg/m^3 density. It is hollowed out to a 1:1000 wall to radius thickness ratio (W:0.001). It masses 1.83*10^-9 kg. When charged negatively, it can support a charge to mass ratio of 1.5 C/kg. Charged positively, it achieves 3 C/kg. 50 keV electrons and 500 keV protons can be deflected. Maximum acceleration is 6.8*10^11 m/s^2.
2) A micrometre-sized macron made up of carbon fibres with 7000 MPa strength and 1790 kg/m^3 density. It is hollowed out to a 1:100 wall to radius thickness ratio (W:0.01). It masses 9.4*10^-16 kg. When charged negatively, it can support a charge to mass ratio of 296 C/kg. Charged positively, it achieves 2960 C/kg. 50 eV electrons and 500 eV protons can be deflected. Maximum acceleration is 5.8*10^12 m/s^2.
3) A micrometre-wide, centimetre-long needle of carbon nanotubes with 63,000 MPa strength and 1000 kg/m^3. Wall to thickness ratio is 1:10 (W:0.01). It masses 7.8*10^-14 kg. When charged negatively, it can support a charge to mass ratio of 356 C/kg. With a positive charge, this becomes 3560 C/kg. 50 eV electrons and 500 eV protons can be deflected. Maximum acceleration (vertical axis) is 9.45*10^13 m/s^2.
4) A 10 nanometre diameter sphere of carbon nanolattice with 200 MPa strength and 300 kg/m^3 density. It masses 1.6*10^-22 kg. When charged negatively, it can support a charge to mass ratio of 8850 C/kg. Charged positively, it achieves 39,665 C/kg. 0.5 eV electrons and 5 eV protons can be deflected. Maximum acceleration is 1*10^11 m/s^2.
We select the maximal value for C/kg depending on the accelerator type. All of these particles are field-emission-limited so they could be further optimized to match their strength-limited C/kg values.
The accelerators we will look at are:
A) A single-stage 10 MV electrostatic accelerator.
B) A 100m long multi-stage electrostatic accelerator with an average acceleration gradient of 3 MV/m, for a final energy of 300 MV.
C) A 100m diameter ring of 10 Tesla field strength.
D) A 100m long electron ‘pushrod’ accelerator.
E) A 100m long proton ‘pushrod’ accelerator.
Their performance is as follows:
A1) 7.7 km/s
A2) 243 km/s
A3) 266 km/s
A4) 890 km/s
B1) 42 km/s
B2) 1,332 km/s
B3) 1,461 km/s
B4) 4,877 km/s
C1) 1.5 km/s
C2) 1,480 km/s
C3) 1,780 km/s
C4) 19,827 km/s
D1) 11,661 km/s
D2) 721 km/s
D4) 35.7 km/s
E1) 11,661 km/s
E2) 10,392 km/s
E4) 913 km/s
Some conclusions can be drawn from these values.
Larger macrons are limited mainly by their strength. The identical D1 and E1 values are because the acceleration reaches the maximum the thin-shelled diamond sphere can handle in the ‘blowpipe’ accelerator.
For the smallest macrons, where C/kg is large, circular accelerators become more interesting than linear accelerators (at least if we are not concerned about the weight of the magnets). We can see the progression of velocities from C1 to C4 being much more pronounced than from B1 to B4.
Also of note is the fact that fibre tubes excel in electrostatic accelerators, but cannot be pushed by a ‘blowpipe’ accelerator as they would bend under those stresses.
Millimetre-sized macrons have an interesting property. They can hold a ‘payload’ of a few milligrams within their hollow core. If they are filled with a fissile material, they can bring it up to sufficient velocities to ignite a nuclear reaction upon impact.
Studies for ignition of ‘micro-fission’ have been conducted by researchers such as Winterberg. They worked out that a 0.2 milligram projectile of uranium-235, when covered in a shell of deuterium/tritium ice, could ignite when compressed to about 10*10^12 Pa.
In response to those calculations, another study was performed where the critical mass of uranium enclosed deuterium/tritium (DT) ice was calculated. It then goes on to describe methods of igniting micro-fission by the impact of hypervelocity projectiles. At 20 km/s, the critical mass is just 0.04 grams.
This mass of uranium would fit inside a sphere of 1.6 mm in diameter. Surrounded by a shell of DT ice of equal thickness, and then a 10 micrometer thick shell of carbon fibre, it would have a total diameter of 3.18 mm and a mass of 0.0422 grams. Average density is 2502 kg/m^3 and the C/kg value when positively charged is 0.0067 C/kg.
At an average voltage gradient of 3 MV/m, a linear electrostatic accelerator would have to be 9.9 km long to push this macron to the required 20 km/s velocity. 10 tesla strength magnets would give it a bend radius of 298.5 km. These are impractical options.
With the shell of carbon fibre handling mechanical stresses, an acceleration of 8.29*10^6 m/s^2 is possible. This means a velocity of 20 km/s is achievable within an accelerator length of 24 meters. About 8.44kJ of energy is consumed in 2.4 milliseconds.
Upon impact, the uranium releases a large portion of its 80 TJ/kg nuclear energy. With 100% burnup, this amounts to a kinetic-to-nuclear energy multiplier of 379,149! Even at a low 10%, this is 37,915 times the energy invested.
A 1 MW accelerator shooting nearly 120 of these micro-fission macrons per second would produce between 38 and 380 GW of power at the target.
The paper also suggests a scaling law where increasing the impact velocity reduces the critical mass by a factor Velocity^12/5.
An impact at 200 km/s could be enough to reduce the critical mass to 0.16 milligrams. This quantity of uranium fits inside a spherical volume that is 0.252 mm wide.
Adding the required layer of deuterium/tritium ice brings the diameter to 0.5 mm and the mass to 0.161 milligrams.
This tiny fuel particle can be comfortably help inside the 1mm diameter diamond shell described earlier as projectile 1. The mass of the macron would increase from 0.00183 milligrams empty to 0.162 milligrams loaded. Its C/kg value and maximum acceleration would fall by a factor 89. An electrostatic accelerator with an average voltage gradient of 3 MV/m would have to be 197.8 km long. A 10 Tesla ring of magnets would have to be 1,186 km in diameter. These are again clearly not wise choices.
In a ‘blowpipe’ type accelerator, an acceleration reduced by a factor 89 still permits a velocity of 200km/s to be achieved within a mere 2.2 meters of length.
The advantage of a smaller, faster micro-fission macron is that it can reach targets further away in less time. However, the energy multiplication effect is greatly reduced.
A 200 km/s projectile contains 20 GJ/kg of kinetic energy. Nuclear fission releases only 4x this amount. If we factor in incomplete burn-up of the fuel, it is likely that there won’t be a significant increase in the energy delivered.
Greater macron velocities can focus more energy into an impact. These impacts generate incredible temperatures and pressures… conditions under which deuterium-tritium fuel with undergo thermonuclear ignition.
Winterberg once again leads the way in providing the theory and maths behind these exciting high energy physics applications. He states that a macron accelerated to over 100 km/s can generate temperatures of over 300 million K and compress fusion fuel to a density of 1000 kg/m^3 (10x that of DT ice).
Another source mentions 100 km/s ignition being possible only if the fuel is surrounded by a collapsing shell of material, and 50 km/s might only needed if the impactor is shaped into a conical shape.
Igniting fusion using hypervelocity impacts has several major advantages.
There is no minimum ‘critical mass’ of fuel, so the smallest macrons can be used. The confinement and fuel burn time depends on the length of the projectile divided by the velocity. This favours elongated fibres with a tip of fuel; also a great shape for achieving 100 km/s with accelerators of reasonable length.
DT fuel contains about 330 TJ/kg of energy, meaning that there is an energy multiplication effect of 66,000 at 100 km/s (assuming 100% burnup) and the energy gain can be maintained at up to 25,690 km/s!
This is an important fact, as many sources mention ignition velocities instead of at least a few thousand km/s.
We could, for example, fill the millimetre-sized diamond shell described above as projectile 1 with DT ice. It would be able to hold 78 micrograms of fuel, which is 43.8 times the shell’s own mass. The macron’s maximum acceleration is reduced by this extra load, but it still manages to reach 100 km/s in a ‘blowpipe’ accelerator of just 31 cm.
A 1 MW accelerator would be firing off 2,500 of these projectiles per second. Up to 64 GW of fusion energy could then be released at the target.
The first obvious application of hypervelocity macrons in space is in propulsion.
The smallest, lightest macrons can be accelerated to velocities exceeding 10,000 km/s. This translates to an Isp of 1,000,000 seconds. Only the most powerful electric engines or advanced propulsion system are capable of this. In the future, fusion energy can exceed this exhaust velocity.
A hypervelocity macron is an improvement over other types of propulsion that achieve those Isps in that it can be shaped out of any sort of common dust, and does not need to emit radioactive material. The energy source could be solar or a closed-cycle nuclear reactor. Kinetic streams of projectiles have been thoroughly discussed as an efficient propulsion system, even up to interstellar velocities.
Fission enhanced macron projectiles are particularly suited for use as a propulsion system. Lower impact velocity requirements means less drive mass, while the ability to greatly reduce the critical amount of uranium also means a great reduction in the minimum pulse energy. Project Orion, for example, assumed a minimum release of 627 gigajoules with each nuclear pulse; any lower and the uranium fuel would be wasted in incomplete burns. Mag-Orion, a Z-pinched variant with a magnetic nozzle, managed 340 GJ thanks to the use of expensive Curium-245. A macron accelerator with 0.04 grams of uranium would only release 3.2 GJ, a hundred times less.
This reduction in pulse energy means smaller suspension, smaller magnetic nozzles or thrust plates, reduced heat loads and smoother accelerations. Less equipment has to be dedicated to recovering and storing energy in between pulses. Maneuvers can be done more precisely. There is much less risk of damage in case of a misfire.
A fusion rocket that uses macron propulsion enjoys many of these advantages too. A kinetic impactor can concentrate the output of an ignition mechanism like an ion beam from a particle accelerator from several milliseconds to less than a nanosecond. The exceedingly difficult peak power needed to ignite fusion is therefore replaced by a thousand to a million times less demanding accelerator. The ignition event can also feasibly be made to take place far away from the engine’s physical structures too. Macrons retain their velocity while drifting through space, so they are just as capable of igniting fusion a hundred meters from a spaceship as a meter away; this is especially important if drive powers on the order of terawatts are needed for a ‘torch drive’.
But these are minor gains compared to the possibility of remotely-accelerated macron-driven propulsion systems.
A stream of macrons can be accelerated a long distance away from a spaceship. They can cross large distances relatively quickly, and then deliver their kinetic energy without any losses. Multiple macrons be fired with a small velocity difference so that over time they bunch together and all simultaneously, providing much higher peak power. All the receiving ship has to do is place an obstacle into the path of the macrons so that the impact creates a plasma explosion. This plasma can be redirected by a magnetic nozzle for thrust. At 100 km/s, the energy needed to vaporize a carbon macron is about 83 times lower than the kinetic energy is contains, so we can expect 1 kg of onboard obstacle material to vaporize 83 kg of incoming carbon macrons.
This sort of propulsion system is similar to beamed propulsion concepts. Propulsive power is delivered without the need for heavy on-board reactors, so high accelerations are possible. The ratio of obstacle material to macron stream mass means that the effective Isp of propellant onboard the spaceship is greatly multiplied too.
Compared to other mass-stream designs for propulsion, each macron carries a tiny amount of energy. It is unlikely that they will do significant damage if a few of them do not hit the intended target.
There is also the possibility to have fission or fusion-enhanced macrons act as propellant for a spaceship. Now we can have small accelerators delivering huge amount of propulsive energy to lightweight spacecraft, to achieve great accelerations and impressive levels of deltaV.
For example, a fission-enhanced stream of macrons could produce a series of nuclear detonations by impacting an obstacle placed inside a magnetic nozzle. The exhaust would be fission fragments with a velocity of 10,000 km/s. A stream of 10 kg/s would release perhaps 80 TW of power. If it takes 1 kg of obstacle material to receive 100 kg of macrons, then a 1000 ton craft with 200 tons of obstacle material would accelerate on average at 1.81g and achieve a total deltaV of 35.7 thousand km/s.
In other words, a 50 GW accelerator can do the work of an 80 TW drive and reduce trip times from Earth to Jupiter to 9.8-14.5 hours for 1000 ton spacecraft.
A laser or particle beam can be used to vaporize incoming macrons. This removes the need for onboard obstacle material to serve as propellant (so deltaV becomes unlimited), but also destroys the structures needed for fission or fusion enhancement, so the energy multiplication effect is also lost.
Those lasers or particle beams can also be used to ‘guide’ the macrons down a specific path. The beams can effectively create an electrical field gradient that holds the macrons in the beam’s centre, or can even be bent by uneven gradients. More details in the Cold Laser-Coupled Particle Beams post.
Another application for hypervelocity projectiles is as weapons.
We have discussed the need for faster projectiles to compensate for the combat ranges imposed by powerful lasers. The ‘solution’ to this need was described as a ‘pellet’ or ‘dust’ gun. Its features accurately describe the properties of a macron accelerator.
At first glance, a macron accelerator produces less watts of output for the same mass of equipment when compared to a laser or a particle beam accelerator. It might also be very bulky as it has many hollow spaces. Worst of all, it does not deliver its energy at or near lightspeed.
A deeper look reveals its advantages.
The macrons have a chance to hit determined by:
Chance to hit = (TR/(0.5*TA * (Distance/MV)^2))^2
Chance to hit is a fraction.
TR is the target’s radius in metres.
TA is the target’s acceleration in m/s^2
Distance is the distance to cross in meters.
MV is the velocity of the macron in m/s.
The chance to hit equation basically compared the cross-section of the target to the area the target could potentially cover in the time it takes for a projectile to arrive.
We can see that a 1000 km/s macron targeting a 5 meter radius spaceship that accelerates at 0.5 g (4.5 m/s^2) from a distance of 2,000 km will hit about 30% of the time.
The equation can be rearranged to determine the effective range of a macron for a certain hit chance:
Effective range = MV * TR^0.5 / (Chance to hit^0.25 * (0.5 * TA)^0.5))
Effective range is in meters.
M is the velocity of the macron in m/s.
TR is the target’s radius in metres.
Chance to hit is a fraction.
TA is the target’s acceleration in m/s^2
If we accept a chance to hit of 10% with a 1000 km/s macron, a 5 meter radius target accelerating at 0.5 g can be engaged at a distance of 2650 km. Note how the range is directly proportional to the macron’s velocity and that increasing acceleration has a much lower effect. Quadrupling the acceleration cuts the target’s deltaV by a factor 4 but only increases the range by a factor 2. This would have to be compensated for by exponentially more propellant.
At the upper end of macron velocities, we can expect effective ranges in the hundreds of thousands of kilometres with good hit chances.
The projectiles deliver their kinetic energy as small plasma explosions that form craters in a target’s armor. An approximation suggested by Luke Campbell is that a hypervelocity impactor excavate a volume of material equal to the kinetic energy divided by three times the yield strength of that material. For something relatively weak, like graphite, this means that the kinetic energy is divided by 2.4*10^8 J/m^3. Vaporizing graphite requires about 500 times more energy per m^3; in other words, a macron accelerator can be 500 times more efficient at removing armor than a continuous beamed weapon, and even better than a pulsed laser.
Stronger materials like carbon fibres require only 5 times more energy to vaporize than to excavate, but this is a still major boost to the destructive efficiency of kinetic weapons.
We can expect a 100 MW stream of macrons impacting hypervelocity to excavate about 0.41 m^3 of graphite or 0.03 m^3 of carbon fibres per second. Using a 100x nuclear energy multiplier upon impact, these volumes can be multiplied to 40 and 3 m^3/s. The actual penetration rate through armor depends on the spacing of the impacts. Closely space impacts, such as within a spot 1 meter wide, would mean a penetration rate of 50.9 m/s through graphite! A wildly maneuvering target might have this potential damage spread out over their entire cross section and then reduced further by the hit chance. Using previous numbers, a 5 meter radius target with a 10% hit chance would find its exposed surfaces ablated at a rate of 5 cm/s if protected by graphite, and 0.38 cm/s using carbon fibres.
Whipple shields could be used to defend against kinetic projectiles. As mentioned in the Propulsion section, very little material is required to destroy an incoming macron. However, the gap created in a whipple shield by the impact of one macron can let through thousands more unimpeded.
It is much harder to add one more layer of shielding material than to fire one more macron to get through it…
The macron accelerator’s offensive performance is also improved by fission or fusion enhancement. It is the only weapon system described so far that output more energy at the target than originally spent. A 1 MW macron accelerator might heavier and bulkier than a 1 MW laser or particle beam, but its actual output can be multiplied a hundred to a thousand times at the target. However much worse the macron accelerator is in a direct comparison, it is more than made up for by the addition of nuclear energy.
To top this all off, macrons are expected to be completely undetectable until they hit their target. Their small size means that they cool down very quickly to temperatures hard to distinguish from background radiation. At the smallest scales (sub-micrometers), even LIDAR cannot interact with them properly, and even if specialized short-wavelength sensors are used, the resolution would be severely limited. If the detection and targeting problems are somehow overcome, the macrons are difficult to destroy en-route. Defensive lasers have very little time to act (seconds at most) and must face the fact that the surface area to mass ratio of the macrons makes them very good at radiating away heat.
For example, a micrometre-sized sphere of carbon fibres could survive a beam intensity of 58 MW/m^2. A micrometre-wide, centimetre-long carbon needle survives 579 GW/m^2. We can therefore expect macrons shaped to handle high laser intensities to defeat even the most power defences (although DT ice is unlikely to remain solid!).
SDI-era research has been done on macron accelerators for space defense. It is now up to you to decide how to make use of hypervelocity macrons.
Thanks for the help of GerritB, Kerr and other ToughSF members for help with researching this topic.