Interstellar Medium Shielding

It might surprise you that you need to shield your ship from the interstellar medium, especially as velocities approach c, the speed of light. This is a result of interstellar space being filled with a diffuse medium of mostly hydrogen, which when relative to a ship at high enough velocities, comes to increasingly resemble ionizing radiation. To boot, the medium also bears a not insignificant component of dust grains, making up 1% of the total mass of the medium on average.

The main dangers are particle-induced heating and erosion from dust grains. Erosion from particles like typical hydrogen atoms is utterly insignificant -- enough that a 1 cm thick carbon shield can go 25,000 light-years (at a speed of 30% of c). However, heating proves to be a significant concern, and erosion from dust grains even more so!

Interstellar Medium Density

To begin with, the interstellar medium density varies greatly, ranging from 10^-4 particles per cubic centimeter in the coronal gas component of the galactic halo of the Milky Way, to 10⁶ particles per cubic centimeter in molecular clouds.

This is important in calculating the flux that the forward portion of the ship will receive at a particular velocity.

Particle Density Table

(In units of particles per cubic centimeter)

Component	Particle Density
Molecular clouds	102-106
H II regions	102-104
Cold neutral medium	20-50
Warm neutral medium	0.2-0.5
Warm ionized medium	0.2-0.5
Coronal gas (Hot ionized medium)	10-4-10-2

The local neighborhood around the sun is assumed to have a particle density of 1 particle per cubic centimeter on average.

Interstellar Medium Composition

By mass, the interstellar medium is 70% hydrogen, 28% helium and 2% heavier elements.
By number of atoms, the interstellar medium is 91% hydrogen, 8.9% helium and 0.1% heavier elements.

There is also a dust component to the interstellar medium; the dust is considerably more dangerous than the diffuse gases as the particles are much larger. Generally, it is considered to be 1% of the total ISM mass in the galaxy. However, in the interstellar medium immediately around the Solar System, the mass of dust is only ~0.5% of the mass of the gas, with the bulk of the particles ranging from 10^-18 to 10^-14 kg. The population of less-numerous but larger particles which pose the greatest hazard is not yet well known.

Erosion from particles

Particle-induced erosion is not taken to be a significant component of the danger in interstellar shielding. For example, with a particle density of one per cubic centimeter at 30% of c a ship's forward shield will encounter around 10¹⁸ ISM particles per square centimeter per light-year traveled.

When a particle impacts a surface, it can knock out particles from the surface in a process known as sputtering. Sputtering simulations suggest that each atom impacting a surface may sputter a few additional atoms. This number will fall off at lower particle speeds but does not significantly increase at higher speeds as the particles largely embed themselves deep in the material so that the do not deposit significantly more energy near the surface. If each impact displaces 2 atoms from the shield, every light year traveled will cause the loss of 2×10¹⁸ atoms per square centimeter. For carbon shields, this is a loss of 40 micrograms per light year per square centimeter, ignoring that not all particles displaced will be lost to space, instead landing back on the shield. This means that a 1 cm thick shield can survive a trip of 56,250 light-years before being worn through.

Note: In the Daedalus report, a number of other mass loss factors and average of a variety of material choices gave a mass loss rate of 80 milligrams per cubic centimeter per light year at a speed of 25% of c.

According to , at a certain velocity regime (5% to 20% of the speed of light), impacting particles may have additional erosive effects by leaving "ion tracks". These tracks are essentially trails of damaged material left in the wake of the ion, which has penetrated deeply into the material. However, there are numerous issues with the paper, according to M. Karlusic's comment , and the ion track effect may not even apply if the shield is made out of conductive metals.

Another paper, , argues that hydrogen and helium atoms at relativistic velocity implant themselves in the material, becoming slowly diffusing gas atoms. These then cause damage through bubble formation, blistering and exfoliation.

However, even in both cases, the erosion is limited to on the order of a millimeter depth every 4 light years travelled (as these papers generally cover the case of a Breakthrough Starshot spacecraft journeying to Proxima Centauri, the closest star). For relatively large starships, these concerns may not matter much. Extrapolating from the rate, for a meter of material to be eroded the starship would need to travel 4,000 light years.

Radiation from particles

At high speeds, particles in the ISM become penetrating radiation. This is covered in the article on space radiation.

Dust Collisions

Interstellar dust grain density ranges from a few hundred to a few thousand grains per cubic kilometer. For the rest of this section, we'll assume a density of 500 grains/km³. This translates to 5×10^-7 grains per cubic meter. Note that the interstellar dust cloud which Earth is moving through, has an order of magnitude higher density: 10^-6 dust grains per cubic meter.

A light year is 9.454 quadrillion meters long, so under this theoretical model there would be ≈ 5×10⁹ dust grains per square meter per light year, colliding with your shield.

As mentioned above, dust makes up about 1% of the mass of the ISM. Under the assumption of one atom per cubic centimeter and with the distribution of atom masses given above, this gives a mass of dust swept out of about ρ_d = 2×10^-7 kg/m^2/ly. The dust density can change considerably between different parts of the galaxy but for our example we will use this value of ρ_d.

Calculating Collision Effects

Note: For collisions, there are various regimes which govern the response of the material after being hit by an impactor. At low velocities (below many kilometers per second), the regime is hydrodynamic. For "hypervelocity" impacts, these matters are entirely governed by the crater regime - in which the impactor leaves a crater in the material. On the extreme end -- the ultra-relativistic regime, impactors are so penetrating that they end up being more like big, mega-bunches of particles leaving cones of primarily radiation and thermal damage.

In the purview of this article, velocities tend to fall between 1% to a hair under c. However, we have entirely no idea of what the intermediate case is like -- when you're already many thousands of km/s, but still below the relativistic regime? We can only guess -- so bear in mind the following is pure and utter conjecture.

Now, dust grain impacts might end up as a hybrid between the hypervelocity crater and the ultra-relativistic "cone". With increasing velocity, the dust grains will penetrate deeper and deeper, along with secondary showers and exotic effects at such high kinetic energies. The original, roughly hemispherical/parabolic crater shape might change to resemble more that of a cone with increasing velocity. This concludes our speculation, and we shall move on to the calculation.

Let's assume that the hypervelocity crater regime still holds for the most part, bearing in mind that up to some high fraction of c, or more generously, when gamma is a large multiple of 1, it will cease to be even slightly accurate.

We can use the method of calculating crater volume that was presented in the article on beam-target interactions: The volume eroded V is related to the energy deposited in the impact event E by

V =	E
	Kh

where K_h is the cavity strength of the material. The cavity strength can be found from commonly reported material properties using K_h = (2/3) K_c × (1 + ln[2 G/K_h]), for compressive strength K_c and shear modulus G. However for approximate uses, K_h tends to be about 3 to 4 times K_c.

The energy of a dust grain with mass m impacted at a relative (and non-relativistic) speed of v is E = ½ m v². If we take a grain at the upper limit of our mass distribution from above, m = 10^-14 kg, and if we are moving at 0.1 c (3×10⁷ m/s), it will have an energy of 4.5 J. At 0.9 c and using relativistic corrections the dust particle energy will be about 1 kJ. An individual dust grain will blast out a hemispherical crater with a radius

r =	(	3 V	)	1/3
		2 π

In ice with a cavity strength of 50×10⁶ Pa, the 0.1 c grain will make a crater 3.5 mm in radius while the 0.9 c grain will make a crater 2.2 cm in radius. As this is a quite large grain, most of the craters will be one to two orders of magnitude smaller and the craters in stronger shielding materials will be orders of magnitude smaller still. You would need to run into an unusually large dust grain to cause a serious problem to your shield.

So instead of individual craters, let's look at the average erosive effect of all the combined craters from the trip. The energy density of the dust particles in the swept volume of the shield will be U_k = ½ ρ_d v². As the dust particles are expected to land randomly from an unbiased uniform distribution across the front of the shield, we can expect that after moving a distance ℓ light years a shield of area A will intercept an energy of E = U_k A ℓ which is more-or-less uniformly spread across the shield. For an erosion depth of d, the eroded volume will be V = d A. We now know from the above relationship of volume to energy deposited that

d A =	ρd v2 A ℓ
	2 Kh

We can now immediately see that the erosion depth per distance traveled is

d	=	ρd v2
ℓ		2 Kh

Relativistic corrections

At small fractions of the speed of light, relativity has a small effect and the results can be approximated by the methods described above. At 0.1 c, for example, non-relativistic formulas are still fairly close. But as speeds get to larger and larger fractions of light speed, relativity becomes more and more important.

To include the effects of relativity, first find the parameters β and γ. β is the fraction of light speed of the motion – if you are moving at 0.1 c your β is 0.1. Once you have β you can find γ = 1/(1 - β²).

With these two parameters, the above formulas can be corrected by substituting in the relativistic result for the kinetic energy E = m (γ - 1) c² for the non-relativistic E = ½ m v². This can be applied directly to the formula for the energy of individual dust grains. For erosion rates, the formula works out to

d	=	ρd (γ - 1) c2
ℓ		Kh

Prospects for Shield Materials

Let's assume we are using ice as a shield. As we will see later, ice remains solid when moving at speeds of approximately 0.1 c or less through the ISM. Using an assumed speed of 0.1 c, our previously assumed ρ_d = 2×10^-7 kg/m^2/ly, and a cavity strength of 50×10⁶ Pa, we find d/ℓ = 1.9 m/ly. Ice thus demonstrates to be a mediocre shielding material at high velocities. Let's look at other materials.

RHA Steel has a cavity strength of 4.7×10⁹ Pa. Using this value, and keeping the speed and dust density the same, we find d/ℓ = 2 cm/ly.

Some kind of perfect carbon nanotube weave could have a cavity strength of around 27×10⁹ Pa, giving d/ℓ = 3.5 mm/ly.

This suggests that for speeds that are not too relativistic and trips that are not too long (less than several hundred light years) a few meters of shielding of some strong material such as steel or advanced carbon allotropes could allow you to reach your destination.

Calculating the Heat Flux

Before we can begin calculating the flux, the mass density of the interstellar medium first be known. The mass density is given by ρ = m n, where m is the mass of the particle and n is the particle density of the interstellar medium.

Since the interstellar medium is not homogeneous, a weighted average must be done per the composition of the interstellar medium. From the section on interstellar medium composition, we find that an atomic weight of 25.8 AMU gives the specified 2% abundance by mass of heavier atoms and the specified 0.1% abundance by number. Performing the weighting, and noting that hydrogen has an atomic weight of 1 AMU and helium of 4 AMU

m = 0.91 × 1 AMU + 0.089 × 4 AMU + 0.001 × 25.8 AMU = 1.292 AMU = 2.145×10^-27 kg

Now we can finally calculate the energy flux with the relativistic flux equation ^[11]

φ = γ ρ β c (γ - 1) c²

where β is the speed of the spacecraft as a fraction of the speed of light, c is the speed of light, and γ is the Lorentz factor

γ = 1/√(1 - β²)

Assuming a particle density of 1 particle per cubic centimeter, at 17% of the speed of light, the flux is comparable to the solar system frost line, enough for ice to begin sublimating. At 80% of the speed of the light, the flux is 52.6 kW/m².

Calculating the Temperature of the Forward Shield

The temperature T of the forward shield is related to the energy flux φ incident on the shield by the Stefan Boltzmann Law

φ = ε σ_SB T ⁴,

where σ_SB = 5.67 × 10^-8 W/m²/K⁴ is the Stephan-Boltzmann constant, and ε is the emissivity of the shield material. The energy flux has already been calculated from the previous section.

Now we rearrange the equation to solve for temperature:

T =	(	φ	)	1/4
		ε σSB

As a surface gets rougher, its emissivity gets closer to 1. Because we expect the surface to get roughed up as it is pockmarked by craters from dust grains, the emissivity of the front surface will generally be higher than the reported emissivities of smooth surfaces of a material.

Conclusions

Below are two tables:

Example Required Shield Thickness Table

(Assuming mean grain mass density of 2×10^-7 kg/m²/ly. Relativistic corrections have been implemented for this table, but still relies on the hypervelocity crater model. Ignores decreases in cavity strength as temperature increases.)

Ship Velocity	Ice (m/ly)	Steel (m/ly)	Perfect Nanotubes (m/ly)
0.1c	1.9	0.020	0.0035
0.2c	Beyond sublimation point, see shield temperature table	0.081	0.014
0.3c		0.19	0.034
0.4c		0.36	0.064
0.5c		0.61	0.11
0.6c		0.98	0.17
0.7c		1.6	0.28
0.8c		2.6	0.47
0.9c		5.1	0.90
0.99c		24	4.3
0.999c			15

Example Shield Temperature Table

Assuming 1 particle per cubic centimeter with an average mass of 1.292 atomic mass units and 1% dust by mass. Ice is assumed to have an emissivity of 0.95 in the infrared. Steel is assumed to have a roughened surface emissivity of 0.7. Graphite is assumed to have a roughened surface emissivity of 0.95. Results are reported to three siginificant figures, which is more than is warranted given the uncertainty in particle density and composition.

Ship Velocity	Interstellar Medium Heat Flux (W/m2)	Ice Temperature (K)	Steel Temperature (K)	Graphite Temperature (K)
0.1c	29.3	153	165	153
0.2c	243	259 Sublimates rapidly	280	259
0.3c	878	357 Molten	386	357
0.4c	2,300		491	454
0.5c	5,160		600	556
0.6c	10,800		723	670
0.7c	22,700		869	805
0.8c	51,400		1070	988
0.9c	154,000		1400	1300
0.99c	2,470,000		2880 Molten	2600
0.999c	27,600,000			4760 Sublimates rapidly

What is clear here is that the interstellar medium flux can present a significant danger at high enough velocities as to sublimate (in the vacuum of space) ice, melt steel, and at ever increasing velocity, even graphite.

Therefore, care must be taken to shield your interstellar spacecraft from the flux if it is moving at a velocity high enough to heat the spacecraft with disastrous consequences.

Additional References

Derivation of the Relativistic Flux Equation

1. The kinetic energy of an amount of mass is given by (γ - 1) m c². To get the power, the kinetic energy is differentiated against time and thus assuming constant velocity, obtain that with ṁ (the mass flow rate); the power is given by (γ - 1) ṁ c².
2. The mass flow is given by the mass per volume encountered every second, doing this in the reference frame of the ship, the ISM density is length contracted to to γ ρ and multiply by A v where A is area.
3. This yields P = γ ρ A v (γ - 1) c², to obtain the flux per unit area divide by A and thereby cancel the A in the earlier expression.

Reference for the Derivation of the Relativistic Flux Equation

Credit

To Tshhmon for writing the article

• To Luke Campbell for helping with particle erosion calculation, editing, and physics and math checking and correction
• To Rocketman1999 for helping with dust erosion distribution
• To Kerr for the relativistic flux equation and derivation