Background noise, cryogenic evaporation and evading detection in space

[MatterBeam]

Hi! This is a post from here.

Permanent and Perfect Stealth in Space

 

Despite the commonly accepted truth in Hard Science Fiction, spacecraft are able to evade detection in space in many circumstances. The Hydrogen Steamer was a design that used liquid hydrogen evaporative cooling to keep a non-reflective surface practically invisible. 

However, it was vulnerable to RADAR and had extremely poor manoeuverability, as it was meant to demonstrate how long it could stay cool. This time, we will design a more advanced, functional and performant stealth spacecraft.

This post builds upon the conclusions drawn from the Stealth in Space is Possible series found here (Part I, II, III and IV). A useful read is the page on the Hydrogen Steamer design.

Detection mechanics

We are considering a large telescope, in space, pointed at a target spacecraft that is very cold, has extremely low reflectivity and is travelling a several kilometers per second. The telescope and the target are separated by a distance of a few dozen to a few million kilometers. To achieve 'stealth', the target must evade detection by the telescope.

The critical question is at what distance the telescope can detect the target.

Previously, we looked solely at the sensitivity of the telescope compared to the intensity of the blackbody emissions from a target at a certain distance. As the distance increases, the inverse square law reduces the intensity of the emissions until they are below the telescope's sensitivity figure. So, for example, a 21 Kelvin target would emit 11 milliwatts per square meter, and a cryogenically-cooled infrared sensor would have a sensitivity of 10^-19W/m^2. By working out the square root of the emissions by the sensitivity, we would get a detection distance - in this case equal to 332 thousand kilometers.

Further research into how telescopes actually work has revealed that this method is unreliable for working out the true detection distances.

The real answer on how far away a stealth spacecraft could be detected actually depends on the relationship between signal strength and noise. Not the usual sense of noise, as in sound you can hear, but noise as all the emissions that a sensor picks up that do not come from a target.

In space, telescopes do not have to deal with atmospheric interference. The sources of noise are instead either internal, such as the thermal photons from hot components, electric resistance in the circuits and quantum inefficiencies in the Charge Coupled Device (CCD), or external, such as sunlight, solar wind, the interstellar medium and the Cosmic Background Radiation. 

Noise threshold can be a million times greater than the actual sensitivity reported.

Many techniques have been developed over the years to improve the performance of telescopes to cut out or minimize the effect of the various sources of noise. These include the use of cryogenic cooling, bandwidth filters, sun-shields, carbon-black coatings, larger collection surfaces, longer observation times and reference sensors to measure deviations. In addition to these physical techniques, digital processing can further improve sensitivity. This is done mainly by subtracting known sources of noise from the final image, to obtain only the difference which can then be attributed to a target's emissions. 

The effect of reducing temperature.

These techniques can be taken to their logical conclusion, resulting in telescopes such as the SPICA-SAFARI proposal. The CCD is cooled to a few milli-Kelvin above absolute zero. The electronics are superconducting, and the optics are also cooled to a handful of Kelvin. This reduces internal sources of noise to near zero. With quantum efficiencies can approach 100%, meaning that every photon collected equals one electron in output, sensitivities on the order of 10^-20W/m^2 and better can be expected. 

However, no amount of cooling can eliminate external sources of noise. Unlike fixed and predictable targets of observation such as a far away star, the background noise cannot be simply be subtracted from the final image, as it might also take with it the emissions of a stealth spacecraft. This creates a 'noise floor' below which a telescope's sensitivity cannot be improved, at least for this task. 
 
Therefore, we must compare the emissions the telescope receives from the target to the levels of external noise it collects to determine at what distance a stealth spacecraft can be detected.

The noise floor

Background noise is a well-documented aspect of astronomical observation. At different wavelengths, certain sources of noise dominate. The types of target we are interested in detecting have very low temperatures. By using this Spectral Calculator, we can work out that their emissions will have wavelengths in the Mid to Far infrared.

Output from Spectral Calc for a 21 Kelvin object. Peak emissions at 138 microns (Far Infrared).

 We can look at this chart to find the dominant source of noise in the Mid to Far Infrared:

The intensity of non-zodiacal light sources, from here.

 

From CIBR.

We see that the Cosmic Infrared Background Radiation dominates in the Mid to Far Infrared, which ranges from 10 to 100 micrometers in wavelength (or 1 to 10 THz in frequency). Its intensity is roughly 10 nanowatts per square meter per steradian.

A depiction of the Cosmic Background Radiation.

A steradian is a measure of solid angle - it is the projection of a two-dimensional angle onto the surface of a three-dimensional sphere. It is a measure of how large an object appears from an observer's point of view. For example, the apparent size of the Sun or the Moon in the sky. It can be converted into the more common unit that is the degree, and its counterpart the square degree. A steradian is equivalent to about 3282 square degrees. The full spherical sky contains 41253 square degrees, so a single steradian represents 7.958% of the entire sky. 

To work out how this translates into a noise floor, we need to calculate how many watts per square meter the telescope receives. This will depend on its field of view. 

Fields of view are listed in arcminutes or arcseconds, which are 1/60 and 1/3600 of a degree respectively. As CCDs are usually square, this actually means an arcminute x arcminute or arcsecond x arcsecond area

We can then convert this field of view into steradian and then multiply it by the background radiation intensity per steradian to find the actual noise floor.

Typical sensors have very small fields of view. The cancelled NASA WFIRST, or 'Wide' Field Infrared Survey Telescope, only had a 2.5 arcsecond field of view, which works out to about half a millionth of a square degree, or about one billionth of a percent of the entire sky. The delayed James Webb Space Telescope has a field of view of 20 arcseconds, but so far most telescopes only have a single arcsecond field of view or less.

We can use this simple relationship:

• Noise floor : FoV * BNI

The noise floor will be in W/m^2.
FoV is the field of view in steradian. 
BNI is the background noise intensity in W/m^2/sr.

A field of view of 1 arcsecond converts into 2.351*10^-11 steradian, which receives a background noise level of 2.35*10^-20 W/m^2 in the 100 micrometer wavelength.

Target emissions

A calculation of the emissions of a blackbody from its temperature and its emissivity, using the Stefan-Boltzman equation, actually gives the total emissions across the entire electromagnetic spectrum, from X-rays to Radio waves. 

This is not a useful measure because as shown below, the emissions of a cold object are bunched around a small range of wavelengths, and all infrared sensors can only detect an even smaller range of wavelengths.

Here is the spectral graph calculator's results for the emissions from a perfect blackbody (emissivity = 1) at a temperature of 30K:

Note that the peak spectral radiance is at a wavelength of 96.59 micrometers (Far Infrared) and that the band radiance, which is the total output per square meter per steradian depends on the 'band' of wavelengths the sensor is picking up. 

High bandwidth therefore allows a CCD sensor to sample the photons from a greater number of different wavelengths, allowing more total signal to be picked up from a target. 

The CCD from the JWST.

However, CCDs have a rather narrow range of wavelengths for which they are tuned for in terms of sensitivity, and a greater number of wavelengths also means more noise from those wavelengths.

Typically, a bandwidth of 1 to 10 micrometers is to be expected. We can approximate the target emissions by using the calculator to find the band radiance 5 micrometers above and below the peak. 

The next step is to calculate the portion of the target's emissions that the telescope intercepts. This is done by working out the solid angle the target occupies in the telescope's view in steradians.

A steradian calculator is handy.

From here.

Solid angle: Cross section area / Distance ^2

A 1m^2 cross section area at 1000m has a solid angle of 1/1000^2 : 10^-6 steradians.

If we also take into account the telescope's collector area, which is the effective area of its main mirror, we can produce the following equation:

• Target emissions received: BR * CSA * TCA / D^2

Target emissions received will be in W/m^2.
BR is the band radiance in W/m^2/sr.
CSA is the Cross Section Area in m^2.
TCA is the telescope collector area in m^2.
D is the distance in m.

Typically, telescopes have many hours to days to repeat their observations of a single spot in the sky, which allows for the collection of a huge number of separate images to be compared for an even greater sensitivity. Data on sensor sensitivity is usually given for 10,000 second observation times for this reason. However, detecting fast spaceships travelling at multiple kilometers per second means that telescopes won't have that luxury - for this reason, we will only consider single-second frames.

Detection equation

From unresolved blob of heat to processed image.

The equations and relationships from above can be combined into a single detection equation that can work out the distance at which a cold stealth spacecraft can be certainly detected. 'Certainly', in this case, is achieved by a signal-to-noise ratio greater than one. In practical terms, this means that the target emissions must exceed the noise floor by a factor of at least 10.

• D: ((BR * CSA * TCA) / (FoV * BNI * SNR))^0.5

D is the detection distance in m.
BR is the band radiance in W/m^2/sr.
CSA is the Cross Section Area in m^2.
TCA is the telescope collector area in m^2.
FoV is the field of view in steradian. 
BNI is the background noise intensity in W/m^2/sr. 

SNR is the signal to noise ratio, at least 10.

Using this equation, in addition to the calculators and information on the background noise, we can establish the shortest distance a stealth craft can approach a telescope without being detected.  

 

For example, a human body (310K, 0.68m^2) would be detected in the Near Infrared (10 micrometers) by a 2m wide telescope using an arcminute field of view and a 10 micrometer bandwidth sensor at a distance of approximately 277 thousand kilometers.

That same telescope would pick up the Hydrogen Steamer design (21K, 7200m^2) in the Far Infrared at a distance of only 21 thousand kilometers.

Using this formula and the previous information, we will look at ways to substantially reduce the detection distance of stealth spacecraft.

ATOMSS: the Advanced Triple Observability Mode Stealth Steamer 

The ATOMSS is a stealth spacecraft design that aims to achieve both extreme levels of undetectability, a greatly extended endurance and a much improved propulsive performance compared to the previously described Hydrogen Steamer design.

It achieves these objectives by using three observability modes (helium, hydrogen, warm), a fully insulated hull, anti-radar structures and super-expansion nozzles for its exhaust. 

The three observability modes make the design more or less visible to telescopes and sensors depending on the situation. 

Schematic for a cryogenic cooling system that uses helium evaporation.

The helium mode guarantees perfect stealth: no sensor will be able to detect it under any circumstance. This mode will be used when penetrating deep into enemy defenses, launching an attack or passing through dense sensor networks. 

The hydrogen mode allows the hull to reach a slightly warmer temperature in situations where stealth is unlikely to be so rigorously needed, such as during an approach to a planet or when passing through a lower-quality sensor network. Hydrogen has better heat absorption properties than helium, allowing for more efficient use of the available coolant mass.  

The 'warm' mode extends a huge area of lightweight, low temperature radiators to handle the spacecraft's waste heat during the long months that interplanetary travel takes. This mode can be maintained indefinitely, without consuming any coolant. 

 

In the original Hydrogen Steamer design, the spacecraft's hull also doubled as propellant tank, so the hull's temperature was that of the propellant. If insulation is used, then the hull's sides can be reduced to extremely low temperatures as they would sit in the shadow cast by the nose. 

The James Webb telescope's sunshield is an example of this use of shadow.

The only detectable cross-section would become the much smaller nose, which is in direct sunlight and so must be actively cooled.

In addition to insulation, the ATOMSS will be equipped with meter-scale structures meant to defeat active detection by high-frequency radar waves. This will give it a bumpy, irregular surface meant to reduce radar returns.

Temperature and pressure drop as the expands.

Finally, the use of nozzles with extreme expansion ratios will allow for high exhaust velocity, high reactor temperature propulsion to be used without leaving a trail of hot or otherwise detectable exhaust. The addition of a pulsed mode with shutters completely eliminates detection by observing the exhaust. The ATOMSS can therefore travel around the solar system just as fast as any other spacecraft.

The details of each of these features will now be described:

-Helium cooling.

Liquid helium cooling is used in specialized electronics.

As made explicit by the detection equation, even the low temperatures achieved by the evaporation of liquid hydrogen are not sufficient to keep a large spacecraft from being detected at tens of thousands of kilometers. 

The only way to achieve even lower temperatures without the use of heavy and energy-expensive heat pumps is to use the evaporation of a fluid that boils at an even lower temperature. In this case, it is liquid helium.

Here is the phase diagram for helium:

From here.

From wikipedia.

The transition from liquid to gas happens at a temperature of 2.17 Kelvin in a vacuum. This is known as the Lambda point of Helium. The phase change from liquid to gas absorbs 20.8 kJ/kg of heat

If a spacecraft's exterior is cooled by liquid helium, it will be practically invisible to any infrared or microwave sensor (peak emissions are in the 1.33 mm wavelength). For example, a 'helium steamer' with 1000m^2 cross-section area and a hull at 2.17 Kelvin, facing a large 5 meter wide microwave telescope with a full 100 micrometer bandwidth, would remain undetectable at a distance of only 43 kilometers!

The main disadvantages of helium are its lower heat of vaporization and heat capacity when compared to hydrogen, so a greater mass of helium is needed to stay cool for the same period. Also, hydrogen can eventually absorb up to 60MJ/kg if heated to a temperature of 3000K, helium will only manage 15.5MJ/kg at that temperature, as its heat capacity is only 5.2kJ/kg/K.

-Vacuum hydrogen

Hydrogen boils at 21K at a pressure of 1 atmosphere. In a vacuum, it instead boils at its triple point of 13.8K. This allows for a substantial reducing in the thermal signature of a stealth spacecraft without having to give up on the incredible heat absorbing properties of hydrogen. 

From this paper.

A 'vacuum hydrogen steamer'of 1000m^2 cross-section area and a hull at 13.8 Kelvin, facing a large 5 meter wide Far Infrared telescope with 10 micrometers bandwidth, would be detected at 6293 kilometers. 

Hydrogen is very interesting when frozen as it can absorb nearly 450kJ/kg when sublimating, and another 14 to 22kJ/kg/K as its temperature increases.

-Warm mode

In this mode, radiators of extremely low mass per area (kg/m^2) would be deployed at a low temperature to remove the few tens of kilowatts that the ATOMSS would generate during the long periods of interplanetary travel. For this mode, we will work backwards from a desired heat rejection performance and a maximum detection distance to find the mass of the radiators dedicated to this mode.

Let us suppose that the ATOMSS needs to get rid of 10 kW of waste heat and that wire radiators are employed. We expect to operate them at low temperatures, so low-density materials and hollow tubing can be used. If the radiator design employed half-empty ultra-high-molecular-weight polyethylene (UHMWPE) tubes a millimeter wide, then it would mass 0.38 grams per meter while having an exposed surface area of 0.00314m^2. A coating of carbon black increases emissivity to near 1. This means that the radiator can dispose of
(5.67*10^-8 * T^4) watts of waste heat per meter of length. 

Only a rectangular cross-section of the wire radiator will be visible to a sensor. This is about 0.001m^2 per meter of wire. 

Together, these factors mean that the radiator will need L: 10000/(5.67*10^-8 * T^4 * 0.00314) meters of wire to dispose of 10kW of waste heat, but it will be detected by a 10m wide telescope at a distance D: ((BR * CSA * 78.5) / 8.46*10^-15))^0.5. We can replace CSA by L*0.001 as it is the exposed cross-section of the wire radiator. This gives a detection distance of D: ((BR * L  * 0.0785) / 8.46*10^-15))^0.5.

At a temperature of 30 Kelvin, the wires will have to be 69,342 km long and will mass 26.35 tons. This gives a detection distance of approximately 25,300 km. 

At a temperature of 50 Kelvin, the wires will be 8,986 km long and mass 3.4 tons. The detection distance increases to 1.03 million km. 

With higher temperatures, such as 100 Kelvin, the radiator mass drops drastically (1478kg) but the detection distance becomes impractical.

A million kilometer detection range might sound very poor, but it is 0.5% of the average distance between Earth and Mars. This means that the ATOMSS can spend up to 99.5% of its travel time in warm mode, without having to expend any coolant. Unless the telescopes are spaced closer than a million km across the Earth-Mars distance (which would result in over a hundred thousand telescopes, more if above-the-plane trajectories need to be covered), then the warm mode is useful as it can increase the spacecraft's endurance significantly. 

-Insulated hull

Vacuum insulation is well-known in the field of cryogenics.

To be used as an open-cycle coolant, the liquid hydrogen or helium must be kept below its boiling point but above its freezing point. While this is a low temperature, it can be at times not low enough to prevent detection.

By using insulation between the propellant tanks and the external hull of a stealth spacecraft, heat transfer is eliminated. This allows the flanks of this design, which are in shadow, the naturally cool down to extremely low temperatures, as low as 2.73 Kelvin, which is indistinguishable from the background temperature of empty space. Insulation can take the form of two or more layers of very reflective and very thin Mylar sheets separated by vacuum that prevent infrared radiation from crossing the gap between them.

Placing propellant tanks in shadow was proposed for the storage of liquid hydrogen.

The nose of a stealth spacecraft is in direct sunlight, and so must be actively cooled. Insulation will not help reduce its temperature, as the main source of heating is external, and the lowest practical temperature is that of the boiling point of whatever coolant (helium or hydrogen) is being used.

An important consequence of this feature is that only the small cross-section area of the nose and perhaps rear will count towards the detection of a stealth spacecraft, since the flanks in the shadow are always too cool to detect.

-Radar countermeasures

Example of RAM applied to a wing's edge.

Radar is a form of active detection, as it relies on a radio signal reaching the target, reflecting off a surface, and then returning to an antenna. Certain techniques and design features can be used to reduce the detectability of the ATOMSS to active detection by radar.

Very short wavelengths, such as millimetric radio or microwaves, can be absorbed by the VANTA-black carbon nanotubes. Longer wavelengths as long as 10 or 100m long can diffract around the spacecraft without interacting with it. The ATOMSS is vulnerable to everything in between: wavelengths of 1cm to 1m (0.3 to 30GHz).

An ideal reflector dish can produce a radio beam up to 70 * Wavelength /Antenna Size degrees wide. 1cm wavelength radio focused by a 10m wide dish would produce a beam width of 0.07 degrees. The beam width can be used to determine the intensity of the radio waves that the target receives, and the return signal spreads again in the other direction. For example, a megawatt radio telescope with a beam width of 0.07 degrees would only produce an intensity of 0.85W/m^2 at a distance of 1000km, which further reduces to 0.72 microwatts per square meter by the time it returns to the antenna. 

This can prevent radio waves from travelling up the exhaust nozzle.

The radio waves then interact with the surface of the stealth craft. Radar-Absorbing Materials (RAM) can be used to absorb between 99.6 and 99.99% of the radio wavelengths between 2.7mm and 10m. 
 

80MHz is 3.7 meter wavelength. 3GHz is 10cm.

 A flat surface reflects 100% of the radio signal back in the direction it came from. A rounded surface spreads the radio waves evenly in all directions. The flanks of a stealth craft are the sides of a cylinder - it causes the sensor signal to be reduced by a factor 1/3.14, compared to a flat surface. The nose of the ATOMSS is rounded, allowing the signal to spread in two dimensions, by a factor 1/6.28.

Using the three effects (beam width, RAM and curved surfaces), the ATOMSS can reliably escape detection by even very powerful radars.

There is more that can be done, such as adding angles to the spacecraft's hull so that the radio waves bounce off in directions that do not return to the sensor platform, but these cannot be relied upon in space because we cannot known where the sensors are, and so attempting to bounce radio waves in one direction might just land them in the antenna of a sensor in an unexpected location. Unlike the ground and sky limiting where stealth aircraft can expect radio waves to appear from, sensor platforms can be placed just about anywhere and can pick up or emit radio waves from any direction.

-Expanded exhaust

Any propellant that is heated by a reactor, such as in a solar or nuclear rocket, needs to be expanded in a nozzle to trade temperature and pressure for exhaust velocity. High expansion ratio nozzle create a flow of exhaust that is at a low temperature, low pressure and near maximal velocity.

An example of a nozzle with a high expansion ratio.

A 'super-expansion' nozzle continues this process to otherwise unreasonable lengths, by creating a cryogenically-cold exhaust at near-vacuum pressure, travelling at the maximum possible velocity. The double benefit to a stealth craft is that they create an undetectable stream of exhaust and increase propulsive performance.

The downside is their size and mass.

From the point of view of a stealth ship, this is an acceptable cost as it would allow them to travel around the Solar System as quickly as any regular military ship. The size and mass penalties are of less consequence to a design that is not supposed to engage in direct combat or in tactical manoeuvers in the first place. 

A further improvement to the super-expansion nozzle is the use of shutters. 

Shutters used in pulsejets.

A simple nozzle is open from both ends. It allows observers from a certain angle to look straight up the nozzle to the throat, where hot gases are emitting a clearly detectable heat signature. A shutter can intermittently block this line of sight by staying closed when propellant is being injected into the nozzle and opening just as the cool, high velocity gases reach the end of the nozzle. Of course, this does impose a pulsed mode of operation.

Example design

We will now work out a sample design for an ATOMSS spacecraft.

As for the Hydrogen Steamer, the mission is to depart from Mars, reach Earth orbit and stay on station several months before returning.

To Scale.

The stealth craft will be 10 meters wide, giving it a frontal cross-section of 78.5m^2

The ends of the ATOMSS are hemispherical, and the rest of the body cylindrical. The nose, which is the end facing the Sun, is cooled to either 2.17, 14 or 15 Kelvin by helium evaporation, hydrogen sublimation or a radiator respectively.

The sort of magneto-calorific cooler mentioned below.

The flanks and end of the ATOMSS are in permanent shadow and kept at 2.17K by a closed-cycle loop with liquid helium. A Peltier-effect or magnetocalorific cooler handles the very low amount of energy absorbed from the exterior through the flanks, from interstellar and interplanetary sources of heat.

The missiles can be dozens of nuclear Casaba Howitzer or Explosively Formed Penetrator warheads.

The mission module can be 200 tons of missiles and 2 tons of sensors. The control module contains the 'brains' of the ship, massing 1 ton and consuming 1kW. A reactor power module is needed to power the electronics when the spaceship is drifting through space and mostly inactive. A 1 ton nuclear reactor and generator producing up to 10kW will be used. We include a further ton of avionics and wiring. An additional 4 tons of cryogenic coolers and conductors is needed. These 209 tons can fit inside a tube 17m long at the back of the spacecraft.

The propulsion module is a high temperature nuclear thermal rocket. Operating at higher temperatures, when coupled with a super-expansion nozzle, allows for maximal exhaust velocity and even more heat to be absorbed per kilogram of hydrogen or helium.

Combining the rocket core with a closed-cycle Brayton turbine and generator might save weight and lead to something like the KANUTER.

The nuclear thermal rocket heats the propellant to a 4000K temperature and ejects the resultant gases at velocities up to 6.5km/s (helium) or 14km/s (hydrogen). All of the rocket's heat is absorbed by the propellant flow. It masses 20 tons and can produce up to 2GW of propulsive power.

The propulsion module is placed at the center of mass of the spacecraft. It can swivel between multiple openings in the hull to apply thrust through the center of mass. This propulsion segment is 5 meters long.

One tank of slush hydrogen at 14 Kelvin containing 400 tons is divided into two segments 30m long.  Another tank containing 888 tons of liquid helium at 2.17 Kelvin is divided into two segments in front and behind the propulsion section, each 39 meters long. Liquid helium can be shifted between these tanks to keep the center of mass in the middle of the propulsion module.

Overall, the ATOMSS is 160m long, giving it a lateral cross-section of 1600m^2.

The hull's insulation, cooling and radar absorbing material adds 6 tons to the craft's mass, while micrometer-thick wire-radiators at 30 Kelvin of the design described above add another 18 tons. 

The total mass is 1548 tons, of which 1288 tons is expendable coolant. A 563 ton drop-tank of liquid hydrogen can be added.
 

Performance and mission capabilities

We will consider the detection distance of the ATOMSS example design against a small telescope (2m wide collector area), a large telescope (10m wide collector area) and a huge 100MW radio telescope (20m wide dish). 

In helium mode, the entirety of the ATOMSS's hull is at 2.17 Kelvin. Liquid helium flows through a heat exchanger in the nose to absorb solar heat, and the helium gas that is produced is pumped through to the reactor and/or engine nozzles. The peak emissions are in the microwave, at 1335 micrometers. A small telescope detects the ATOMSS from the front at 4.25 km, and from the side at 13.6km. A large telescope only improves these distances to 20.9 km and 67.9km respectively. 

However, in Earth orbit, the ATOMSS can only use the helium mode for 2 days. This is because the nose absorbs 106.8kW of sunlight, which requires 5.14 kg of helium to be vaporized per second.

In hydrogen mode, helium gas is circulated from the nose to the slush hydrogen tanks. This vaporizes the hydrogen and cools down the helium gas, effectively transferring heat from the VANTA-black on the nose to the internal heatsink. The ATOMSS' flanks remain at 2.17 Kelvin but the nose warms to 14 Kelvin as this is the temperature the hydrogen evaporates at. It emits in the 205 micrometer range, which is the Far Infrared. A small telescope finds the nose at 2000km, and a large telescope detects it at 10,001 km.

The hydrogen mode can be maintained for up to 20 days. 

Using the open-cycle cooling reduces the deltaV capacity of the ATOMSS craft, so they are restricted to situations where 'good' or 'perfect' stealth is required.

In the warm mode, the ATOMSS extends 23.16 billion kilometers of micrometer-thick hollow wires, with a total emitting area of 74.4 million square meters. These wires radiate at 15 Kelvin, with peak emissions in the hundreds of micrometers.

Blackbody radiation is emitted at random polarization. This means that the radiator wires are transparent to perpendicular wavelengths, but will absorb parallel wavelengths. This means that a forest of microwires spaced by 10 micrometers and placed like hairs all over the hull will allow 50% of the heat to escape, regardless of inter-reflection rules.

The 'hairs' extend 0.43 meters from the hull. This means that a telescope will only see, at most, 1608m^2 of radiator.

In warm mode, the ATOMSS can be detected by a small telescope at 161 thousand km and a large telescope at 838 thousand km.

The huge radar telescope does not care which mode the ATOMSS is in. The 100MW radar emitter, if producing 1m long radio waves, can be focused into a 0.7 degree wide beam. In the best case scenario, the 100MW radio beam reflects off the nose of the spacecraft. In the worst case, it catches the flat side.

Waves reflected off the spacecraft's hull are weakened by a factor 1/(1.375 * 10^-8 * Distance^4) by propagation, a factor 1/10000 by absorption and by a factor 78.5/6.28 for the nose and 160/3.14 for the flanks. This means that a 100MW signal returns to the 314m^2 dish as 2.58*10^15/D^4 W/m^2 from the nose and 1.16*10^16/D^4 W/m^2 from the flanks. 

If we take into account the 10^-12 W/m^2/sr background noise and want a 10:1 signal to noise ratio, we find out that the nose is detected at a distance of 4007 km and the flanks at 5835 km.

A DeltaV chart that is more useful for spacecraft that stay in space.

The ATOMSS can depart from an orbit near Phobos, which is Mars's largest moon and a likely construction site for space warships, and place itself at the edge of Earth's Sphere of Influence (an altitude of 924,000km) about 8.5 months later with just 4.34km/s of deltaV. The departure and insertion burns consume all of the liquid hydrogen in the drop tank.

Stealth can be maintained in this orbit indefinitely in the warm mode, or for up to 20 days in the hydrogen mode if safety against detection by large telescopes is required. If the stealth ship is detected or needs to approach an enemy craft very closely, it employs the helium mode which guarantees perfect Infrared stealth. In the helium mode, about 5.14kg of liquid helium is vaporized per second. This can be fed to the nuclear thermal rocket to produce a thrust of 35kN, which is enough to accelerate the ship from of 22 to 134 mm/s^2 without any waste. More helium or hydrogen can be consumed to perform escape manoeuvers with the full 2 GW output of the main engine, at an acceleration of 185 mm/s^2 to 2.3 m/s^2.  

To perform an attack, it simply releases the weapons it is carrying. They can deorbit themselves and crash down into targets in lower orbits at a relative velocity that can reach 10.7km/s, in addition to whatever propulsion they may have that increases this number.  

After the mission is completed, the ATOMSS can return to Mars by using another 4.3km/s return trajectory and consuming the last 93.5 tons of hydrogen (if the ammunition has been expended). 

 

Decoys and Laser Jamming

Because of the way the stealth ship is detectable, decoys and laser jamming can be used to great effect.

How an Infrared sensor sees flares.

For example, the front of the ATOMSS spaceship can be reproduced by a 4m wide sphere of graphite cooled by a small tank of liquid hydrogen. To an infrared sensor or a radio telescope, this is indistinguishable from the front of a stealth craft. The decoy cannot be distinguished from the real craft when accelerating to move, because the exhaust is undetectable. 

Radar jamming.

Many of the techniques used to prevent radar from used to locate the ATOMSS become incredibly effective in space, due to the inverse square law allowing a tiny emitter on the stealth ship to overpower or confuse a huge and powerful emitter on the radio telescope. Electronic warfare against radars can be relied upon.

Another method of jamming is to use lasers. Cold objects like a stealth steamer are only detectable by trying to pick up their emissions in the rather narrow back of wavelengths that they emit in. For example, the emissions peak of a 14 Kelvin object is between 140 to 320 micrometers. The intensity of these emissions fall by a factor 100 or more outside of this peak. 

This means that a small number of lasers that can reproduce the emissions peak of a cold object while producing only a few watts can render any sensor looking at these wavelengths useless as the signals from the ATOMSS are drowned out by the signals from these lasers.  

 

Conclusion

Stealth steamers can stay permanently undetectable at reasonable distances. With the use of solid or slush hydrogen, they can maintain stealth at close ranges for months on end with relatively small supplies of the coolant. The addition of a helium mode makes them perfectly invisible and able to escape tough spots or come within a stone's throw of any target.


With super-expansion nozzles, the stealth ship can be manoeuverable and travel great distances at low to moderate accelerations. 

The example worked out in this post uses technologies that can be considered 'near-future'; not much better than what is available today.

'Far-future' technologies can carry this concept to greater heights of performance. Gas-core nuclear rockets, for example, can significantly increase the acceleration under stealth. Superconducting electric motors can allow for lightweight and efficient heat pumps that allow for convenient helium-mode stealth without the need for helium. Extremely long carbon nanotubes that can be layered on top of each other allows the carbon-black coating to start absorbing some of the shorter radio wavelengths and reduce the stealth ship's radar returns...