Calculates the expected emission spectra for a dyson sphere of various size.

dyson_calcs.py

from math import pi

"""
Licensed by Vyryn under the MIT License.

This script simplistically calculates emission temperature and peak emission wavelength for a Dyson sphere of varying
 parameters. 


The US energy sector is ~$350B while US GDP is ~20T
That's 1.75%, for simplicity let's say 2%.
We can perhaps scale up to a solar system economy and assume the same portion of resources spent on energy 
production for a first order approximation of what that might be. I'll assume our solar system composition is 
average and *no* stellar engineering/starlifting - which is likely pessimistic - but harvesting of 50% of the 
system's planetary mass.

Available mass: 1.9E27 kg Jupiter, all other planets combined are another 40% on top of that, so 2.66E27 kg. 2% of 
50% of that is 2.7E25 kg. A K2 would probably want to use less, but for a mature K2.7+ this seems fair.

So 2.7E25 kg is perhaps a reasonable estimate of the per-solar system energy mass budget of a ~K3 civ. Yeah, 
insane sentence but hey. 

We also need a thickness, we'll vary that and show results for a variety because it turns out to be the main deciding
 factor.

For density, I have no clue what's resonable beyong it likely being somewhere between silicon (one of the lightest 
materials with desirable properties @2.3) and tungsten (one of the densest materials with desirable properties @19.3)
"""
# Some constants
sigma = 5.670_374_419E-8
b_prop = 2.897_771_955E-3
cmb = 2.7  # Cosmic microwave background temperature, in K. this is the best possible heat sink


# Parameters you can adjust
mass_avail = 1.9E27 * 1.4  # Jupiter's mass in kg, * 1.4 for the rest of the planets
mass_fraction_exploited = 0.1  # The fraction of potentially available mass exploited by the civilization
fraction_for_energy = 0.02  # The fraction of that mass used for energy production
# thickness = 0.01  # Thickness of the Dyson sphere. (Defined below in a range as this is more or less the independent 
# variable)
civ_eff = 1  # The achieved efficiency as a fraction of the thermodynamically optimal carnot efficiency
sun_radius = 696_340_000  # meters
sun_temp = 5778  # Kelvin

mass_used_for_energy = mass_avail * fraction_for_energy * mass_fraction_exploited
print(
    f'With an available mass of {mass_avail}kg and {fraction_for_energy * mass_fraction_exploited * 100}% of that used '
    f'for energy production, {mass_used_for_energy}kg is available.')


def volume_from_t_and_r(t, r_):
    """
    Of an optimal Dyson Sphere.
    """
    a = r_ + t / 2
    b = r_ - t / 2
    v = pi * 4 / 3 * (a ** 3 - b ** 3)
    return v


def mass_from_density_and_vol(d, v):
    """
    Of an optimal Dyson Sphere
    """
    m = d * v
    return m


def vol_from_mass_and_density(m, d):
    """
    Of an optimal Dyson Sphere.
    """
    v = m / d
    return v


def radius_from_t_and_v(t, v):
    """
    Gives the possible radius of a dyson sphere from the thickness and volume it occupies.
    """
    # v/pi*3/4 = (r+t/2) ** 3 - (r-t/2) ** 3
    nominator = (3 * v - pi * t ** 3) ** 0.5
    denominator = 2 * (3 * pi) ** 0.5 * t ** 0.5
    return nominator / denominator


def g_cm_3_to_kg_m_3(density):
    return density * 1000


def meters_to_au(dist):
    return dist * 6.685E-12


def au_to_meters(dist):
    return dist / 6.685E-12


def radius_from_mass_density_and_thickness(m, d, t, au=False):
    """
    The name is pretty self-explanatory.
    """
    v = vol_from_mass_and_density(m, g_cm_3_to_kg_m_3(d))
    radius = radius_from_t_and_v(t, v)
    if au:
        return meters_to_au(radius)
    return radius


def carnot(h, c=cmb, eff=civ_eff):
    """
    The theoretically achievable heat pump efficiency with the given hot and cold temperatures. eff is assumed to be 
    1 by default but could be less, you can set that above.
    """
    if c < cmb:
        c = cmb
    if h > c:
        return (1 - c / h) * eff
    else:
        return (1 - h / c) * eff


def round_(val, places):
    """
    A helper method that rounds to the specified number of places after the decimal point.
    """
    return round(val * 10 ** places) / 10 ** places


def best_radiative_eff(radius, temp, cmb_=cmb):
    """
    The ideal radiator at distance radius from the sun and solar temperature temp.
    """
    sa = 4 * pi * radius ** 2
    waste_t = (temp ** 4 / sa * 1/civ_eff) ** (1 / 4)
    if waste_t > temp:
        return temp
    if waste_t < cmb_:
        return cmb_
    return waste_t


def t_at_dist_and_lum(d_, au=True):
    """
    Temperature at distance and luminosity. AU is whether the units are in AU (default) or m (SI).
    """
    if au:
        d_ = au_to_meters(d_)
    t = sun_temp * (sun_radius/2*d_) ** 0.5
    # t = (luminosity_ / (16 * pi * sigma * d_ ** 2)) ** (1 / 4)
    return t


def peak_black_body_wavelength(temp_):
    """
    The peak emission wavelength at the given black body temperature.
    """
    lambda_peak = b_prop / temp_
    return lambda_peak * 10**9  # nm


def spectra(wavelength):
    """
    Converts a wavelength in nanometers to a spectrum name.
    """
    if wavelength > 1E8:
        return 'Radio'
    elif wavelength > 1E6:
        return 'Microwave'
    elif wavelength > 10_000:
        return 'Far IR'
    elif wavelength > 2_500:
        return 'Mid IR'
    elif wavelength > 750:
        return 'Near IR'
    elif wavelength > 380:
        return 'Visible'
    elif wavelength > 315:
        return 'UV-A'
    elif wavelength > 280:
        return 'UV-B'
    elif wavelength > 100:
        return 'UV-C'
    elif wavelength > 10:
        return 'EUV'
    elif wavelength > 10:
        return 'EUV'
    elif wavelength > 0.2:
        return 'Soft X-Ray'
    elif wavelength > 0.01:
        return 'Hard X-Ray'
    else:
        return 'Gamma Ray'


def results(mass_, thickness_, density_, t_sun=sun_temp):
    """
    A "Make it pretty" printout summary at the given parameters.
    """
    d = radius_from_mass_density_and_thickness(mass_, density_, thickness_, au=True)
    t = t_at_dist_and_lum(d)
    t_waste = best_radiative_eff(d, t_sun)
    eff = carnot(t, c=t_waste) * 100
    emission_lambda = peak_black_body_wavelength(t_waste)
    prefix = ''
    if thickness_ < 1E-5:
        thickness_ *= 10E6
        thickness_ = round_(thickness_, 3)
        prefix = 'n'
    print(f'With a {thickness_} {prefix}m thick shell at {density_} g/cm^3 density, the shell is'
          f' {round_(d, 3)} AU distant.'
          f' Temp is {round_(t_waste, 2)}K. Carnot efficiency would be '
          f'{round_(carnot(t_sun, c=t_waste) * 100, 2)}% and peak '
          f'emissions at the {round(emission_lambda, 2)}nm ({spectra(emission_lambda)}) wavelength')


def show_silicon_and_tungsten_radii(mass_, thickness_):
    """
    Simply a helper method to display both silicon and tungsten for the given mass and thickness.
    """
    results(mass_, thickness_, 2.3)
    results(mass_, thickness_, 19.3)

thickness = 0.1
for i in range(11):
    show_silicon_and_tungsten_radii(mass_used_for_energy, thickness)
    thickness /= 10