Interstellar Medium Shielding

It might surprise you that you need to shield your ship from the interstellar medium, especially as velocities approach c. 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 regime 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. In the interstellar medium immediately around the Solar System, the mass of dust is ~0.5% of the mass of the gas, with the bulk of the particles ranging from 1E-18 to 1E-14 kg; however, 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, at 30% of c a ship's forward shield will encounter 1E+18 ISM particles per square centimeter per light-year traveled (ignoring differences in ISM density through the journey).
A light year contains 946.1 quadrillion centimeters. In that length, there are thus 946.1 quadrillion cubic centimeters, and assuming a particle density of one per cubic centimeter, there are 9.467E+17 particles in that volume, rounding up to 1E+18.

If each impact displaces 2 atoms from the shield, every light year traveled will cause the loss of 2E+18 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.

To get the mass loss rate, 2E+18 times the atomic mass of carbon gives 40 micrograms.

This means that a 1 cm thick shield, can survive a trip of 56,250 light-years before being worn through. At high relativistic velocities however, space-time contraction is significant enough that the effective ISM density increases.

The density of carbon, times the length, divided by mass loss rate gives the max trip length due to erosion.

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.

Dust Collisions

!	Notice:
	This section is currently a work in progress and information here may not be correct.

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 $5\cdot 10^{-13}$ grains per cubic centimeter. Note that the interstellar dust cloud which Earth is moving through, has an order of magnitude higher density -- $10^{-12}$ dust grains per cubic centimeter.

As was mentioned in the erosion section, a light year contains 946.1 quadrillion cubic centimeters, so for our ship there would be 473,050 dust grains per square centimeter per light year.

According to , the mean mass of a dust grain is $3\cdot 10^{-16}$ kilograms.

Calculating Collision Effects

An approximation for calculating the excavated volume from a dust grain impact is:

$V_{exc}\approx {\frac {E_{k}}{3\cdot \sigma Y}}$

• where $V_{exc}$ is the volume excavated by the impact
• where $E_{k}$ is the kinetic energy of the dust grain
• where $\sigma Y$ is the yield strength of the material being impacted.

If we approximate the crater as a hemisphere, we can calculate the depth:

$D_{c}\approx {\sqrt[{3}]{\frac {0.159\cdot E_{k}}{\sigma Y}}}$

• where $D_{c}$ is the depth of the impact crater.

The yield strength is the amount of tensile stress you can put on a material until it permanently deforms.
Note: Some materials do not have a yield strength that is below their ultimate tensile strength -- in which case, you may substitute the earlier for the latter.

Prospects for Shield Materials

Despite the relatively low kinetic energies of these grains contrary to popular discussions of dust impacts -- fanciful imaginations conjuring up multi-tonne TNT explosions on the shield, things do not look great. Although the yield strength of ice isn't particularly well measured, it should be roughly 0.1 MPa. Given this, and a velocity of 0.1 c (the velocity for which ice is still a solid, see latter sections):

2,840 meters $\approx {\sqrt[{3}]{\frac {0.159\cdot 0.136J}{0.1MPa}}}\cdot 493050$

This is the cumulative length eroded by 493,050 crater impacts on the ice shield in a light year.

For our very best modern carbon fiber, the prospects look significantly better, but still bad:

68.89 meters $\approx {\sqrt[{3}]{\frac {0.159\cdot 0.136J}{7GPa}}}\cdot 493050$

Of course, we are making oversimplifying assumptions like the crater ejecta not landing back on the shield, a point raised in the section about particle erosion. Starships may prefer a combination of approaches, rather than just relying on their shield to passively handle the dust-induced erosion -- otherwise, the shield will grow unreasonably thick: for a trip of 4 light years at 0.1c (like to Proxima Centauri), the carbon fiber shield is already at 280 meters. For Tau Ceti, it's 827 meters.

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:

$\rho =mp$

• where $\rho$ is the mass density of the interstellar medium
• where $m$ is the mass of the particle
• where $p$ 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. We can assume that all of the heavier elements are iron atoms as an approximation.

$1.475\cdot 10^{-27}$ kg (average mass) = $(0.7\cdot H+0.28\cdot He+0.015\cdot Fe)/3$

• $H,\,\,He,\,\,Fe$ respectively refer to the atomic masses of hydrogen, helium and iron.

Now we can finally calculate the flux with the relativistic flux equation ^[7]:

$\phi =\gamma \rho vc(\gamma -1)c^{2}$

• where $\phi$ is the interstellar medium flux
• where $\rho$ is the mass density of the interstellar medium
• where $v$ is the velocity of the ship
• where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} is the speed of light
• where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} is gamma, calculated with:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = 1/\sqrt{1-v^2}}

Assuming a particle density of 1 particle per cubic centimeter, at 41% of the speed of light, the flux is comparable to what Earth receives from the sun, already enough for ice to begin melting. At 80% of the speed of the light, the flux is 35,327 W/m².

Calculating the Temperature of the Forward Shield

The temperature of the forward portion is given by the Stefan Boltzmann Law :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_e = A_d \epsilon \sigma_{sb} T^4 }

• where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_e } is the radiant power
• where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_d} is the radiating/absorbing surface area
• where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} is the emissivity of the radiating/absorbing material
• where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_{sb}} is the stefan boltzmann constant
• where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is the temperature of the material

Now we rearrange the equation to solve for temperature:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T= \sqrt[4]{\phi_e} / (\sqrt[4]{A_d} \sqrt[4]{\epsilon} \sqrt[4]{\sigma_{sb}})}

Before we can solve the equation for temperature, the radiant power must be obtained from the interstellar medium flux, given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_e = IA}

• where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is the interstellar medium flux
• where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is the area exposed to the interstellar medium flux

Conclusions

A calculator for interstellar medium shielding is provided here:
[link?]

Below are three tables:

Grain Kinetic Energy per Velocity Table

(Assuming mean mass of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 \cdot 10^{-16}} kilograms.)

Ship Velocity	Dust Grain Kinetic Energy
0.1c	0.136 J
0.2c	0.556 J
0.3c	1.302 J
0.4c	2.456 J
0.5c	4.171 J
0.6c	6.741 J
0.7c	10.793 J
0.8c	17.975 J
0.9c	34.894 J
0.99c	164.17 J
0.999c	576.09 J

Flux per Velocity Table

(assuming particle density of 1 particle per cubic centimeter)

Ship Velocity	Interstellar Medium Heat Flux
0.1c	20.122 W/m2
0.2c	167.283 W/m2
0.3c	603.483 W/m2
0.4c	1579.947 W/m2
0.5c	3549.648 W/m2
0.6c	7451.7 W/m2
0.7c	15,593.049 W/m2
0.8c	35,326.58 W/m2
0.9c	106,195.696 W/m2
0.99c	1,698,225.484 W/m2
0.999c	18,973,260.691 W/m2

Example Shield Temperature Table

Assuming a cylindrical shape, radius of 10 meters and thickness of 1 meter

Ship Velocity	Interstellar Medium Heat Flux	Ice Temperature	Graphite Temperature
0.1c	20.122 W/m2	113.558 K	123.207 K
0.2c	167.283 W/m2	192.824 K Beyond sublimation point	209.208 K
0.3c	603.483 W/m2	265.744 K	288.325 K
0.4c	1579.947 W/m2	338.033 K Too hot even at standard pressure	366.756 K
0.5c	3549.648 W/m2		449.017 K
0.6c	7451.7 W/2		540.481 K
0.7c	15,593.049 W/m2		650.054 K
0.8c	35,326.58 W/m2		797.521 K
0.9c	106,195.696 W/m2		1050.131 K
0.99c	1,698,225.484 W/m2		2099.982 K
0.999c	18,973,260.691 W/m2		3839.303 K Beyond sublimation point

Notes: Ice has an emissivity of 0.97, while Graphite has an emissivity of 0.7

The parameters vary with changing exposed area, area and emissivity, flux. 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, 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\gamma -1)mc^2} . To get the power, the kinetic energy is differentiated against time and thus assuming constant velocity, obtain that with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{m}} (the mass flow rate); the power is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\gamma -1)\dot{m} c^2} .
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma \rho} and multiply by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Av} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is area.
3. This yields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \gamma \rho Av(\gamma -1)c^2} , to obtain the flux per unit area divide by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and thereby cancel the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} in the earlier expression.

Reference for the Derivation of the Relativistic Flux Equation

Credit

To Tshhmon for writing the article

• To lwcamp for helping with erosion calculation
• To Kerr for the relativistic flux equation and derivation