Human Space Populations

How much living matter (biomass) can the Solar System support? The most readily accessed resources are carbonaceous asteroids and maybe comets. We therefore measured the amounts of organic carbon and of mineral nutrients in asteroid/meteorite materials, in soluble bioavailable forms and as total contents. We also know how much of elements are needed to form biomass, and how much of such resource materials are present in the Solar System. asteroids and in comets. From these data we can estimate the amounts of biomass and populations that can be sustained by space resources.

Based on the limiting elements N and P, water-extractable materials in one kilogram of carbonaceous asteroid soils can support 0.6 grams of biomass. On this basis, bioavailabe extractable materials in the 1e22 kg carbonaceous asteroids can support 6e18 (six million trillion) kilograms of biomass, six thousand times more than the biomass presently on the Earth, that supports six billion humans. The extractable asteroid materials could then support on the order of 40e12 (40 trillion) humans. Using the total elemental contents of the carbonaceous asteroids could support a biomass and population a hundred times larger yet, 4,000 trillion humans, comparable to the population of a million Earths. Materials in the comets could support biomass and populations even ten thousand times larger, comparable to ten billion Earths, in our Solar System alone. Billions of similar solar systems throughout the galaxy can support amounts of life and human populations billions of times still larger.

Further details are presented in the paper below and in the references therein.
 

Links

Inquiries about space resources and populations and the Society for Life in Space (SOLIS): info@solis1.com See also www.astro-ecology.com and  www.panspermia-society.com More details about seeding other solar system (Directed Panspermia), astroecology and astroethics: “Seeding the Universe with Life: Securing Our Cosmological Future”    Home

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Technical Paper

Life in the Cosmological Future: Resources, Biomass and Populations, M. N. Mautner, Journal of the British Interplanetary Society 2005, 59, 167-180.  m.mautner@solis1.com

Overview
 

Our panspermia missions, and human expansion in space, can seed the galaxy with life. What are the prospects for life as it expands into the future?

            Life is a self-propagating process of structured matter, and this process always requires a flow of matter and energy. The rules of ecology are universal. In order to predict the future scope of life, we can form some projections of future resources of matter and energy. 

            We can assume that life will derive energy from the stars. Some of the present stars may last for trillions of years, and more stars will continue to form. We can estimate their energy output through the coming eons. We also know how much energy biomass requires. Combining these data, we can estimate the biomass that can be sustained and how long life can last, about future stars. The numbers are immense.

            Future cosmology may be determined by black holes, dark energy and dark matter, whose behavior is not well understood yet. However, our knowledge is growing and we can already apply astroecology considerations to future life and its resources.

            We can predict the future scope of life in the galaxy and in the universe using current cosmology. This projected future is immense: up to 10³⁴ kg-years of life in the Solar System, 10⁴⁶ kg/years about stars in the galaxy, and up to 10⁵⁹ kg-years in the universe if all matter was turned into biomass. We can express the potential biomass life in numbers, but these amounts, and the richness of life that it can create, are beyond comprehension. The potential scope of future life can strengthen our purpose to expand life in the universe.

Populations Around Red Giant and White Dwarf Suns

Past observations of the universe cover only 14 billion years. The detailed evolution of the Sun, the galaxy and the universe for trillions of years to come can be predicted only by unverified theories. Little research has been done even in theory about the detailed future of the Solar System and its resources, or on the future of other stars and their planets on long time-scales. This section will rely on the cosmological model presented by Adams and Laughlin (1999). Apart from resources, another important factor in the long term is wastage. Recognizing that wastage may severely limit the amount of living matter, future societies with advanced technologies may prevent wastage completely, and this will be assumed in the following discussion.

After the current phase, the Sun will become a red giant and then a white dwarf star. The Earth will be destroyed during the red giant phase but the Solar System itself may remain habitable much longer, even by contemporary humans, during the 10²⁰ year white dwarf phase of the Sun (Adams and Laughlin, 1999). The population will only need to move closer or further from the Sun as its luminosity varies. Adjustments to the orbits of space colonies and of free-living humans in space as the Sun's luminosity changes should be simple. The other estimated 10¹² white dwarfs in the galaxy may similarly sustain life for 10²⁰ years.

During the red giant phase the luminosity of the Sun will increase on the order of 2,000 times compared with its current value. This phase will end in an unstable period accompanied by thermal pulses, when the luminosity may reach about 6,000 times its current value (Ribicky and Denis, 2001). Can life find habitable zones in the Solar System during this period?

The habitable zone about a star is defined by the temperatures at which life can survive. For example, it may be defined as the zone where the equilibrium temperature of an object is between 0 and 100 degrees centigrade where liquid water can exist under the pressure of one atmosphere. We may also define a "comfort zone" about the distance from the Sun where the equilibrium temperature is a mild 25 degrees centigrade.

The temperature of an object in orbit about the Sun is determined by equilibrium between the absorption of solar radiation and the emission of thermal radiation. The equilibrium temperature increases with the luminosity of the Sun and it decreases with the heliocentric distance, as shown by equation (A14) in the Appendix 2.3. When the luminosity of the Sun increases to 2,000 times its present value, the habitable zone will be at a distance between 21 and 39 au, and the comfort zone of 25 C will be at about 33 au, about the distance of Neptune. At a later stage when the luminosity of the Sun increases to 6,000 times of its present value, the habitable zone will be between 36 au, (i.e., the distance from the sun to Pluto) and 68 au.  The comfort zone will be about 57 au. By this time the Sun will have lost some of its mass, reducing its gravitational pull, and the planets will move out further (Ribicky and Denis, 2001). Neptune, Pluto and the inner Kuiper Belt comets may move to orbits closer to or inside the habitable zones. Space colonies or free-sailing humans can also move to these zones.

The location of the habitable zone will force populations to move to the area of the Kuiper Belt where there are major resources. There are an estimated 35,000 Kuiper Belt objects including Pluto with radii larger than 100 km (Lewis, 1997). The total mass of these comets is at least 10²⁴ kg, a hundred times more than the mass of the carbonaceous asteroids. These cometary nuclei are rich in water, organics and inorganic nutrients. The biomass that they can support using all of their elemental contents is 6x10²² kg, or in round numbers on the order of 10²³ kg. A population can live on these resources for a period on the order of a billion years during the red giant phase of the Sun, giving an integrated BIOTA_int on the order of 10³² kg-years. In comparison, the amount of life that has existed to the present may be estimated as 10²⁴ kg-years.

The Kuiper Belt resources will be available only if these cometary nuclei survive the red giant phase of the Sun. Pluto and the other Kuiper Belt objects contain various ices that could evaporate if heated. However, objects that remain below 150 K may not evaporate except for losing some highly volatile substances from their surfaces. This process forms a non-volatile protective crust that prevents further losses, as found in burnt-out comets. Even at the hottest stage of the Sun, objects further than 450 au will remain at these low temperatures. This applies to the outer Kuiper Belt that may extend to 1,000 au. Of course the Oort cloud comets at 40,000 au will be much colder, at 11 K even when the Sun is the most luminous. If most of the Kuiper Belt and the Oort Cloud comets are preserved, the mass will be on the order of 10²⁴ - 10²⁶ kg including organics and water.

Can these resources be accessed by humans? A velocity of 10^-4 c can be reached by current solar sails, and this speed is also similar to the orbital speed of the Earth and other objects. Space travel at this speed to the comfort zone at 100 au when the Sun will be most luminous will last only 16 years, and travel to 1000 au to collect resources of the Kuiper Belt will last 160 years. Such travel times are accessible for humans with natural or moderately extended life spans. It is comforting that human populations may survive in the Solar System in this manner through its hottest days, since alternatives such as travel to the colder Oort cloud at 40,000 au would last over 6,000 years and interstellar travel to nearby habitable stars may last up to millions of years, and their feasibility for humans is uncertain (Mauldin, 1992).

Once past the red giant period, biological life, possibly including human life, may continue in this Solar System for up to 10²⁰ years using the power output of the white dwarf Sun. It was estimated that at this stage the Sun will be reduced to about the size of the Earth with a surface temperature of 63 K and a luminosity of 10¹⁵ Watts (compared with the Sun’s current luminosity 3.8x10²⁶ Watts), that will be powered by the capture and annihilation of dark matter, if this speculative process actually occurs (Adams and Laughlin, 1999). Populations can then move close to the white dwarf Sun and capture its power in a Dyson Sphere (Dyson, 1979b and 1988) as suggested by Adams and Laughlin (1999). It is possible that radiation may be focussed by mirrors or converted to electrical energy for heating to create biologically habitable environments, although at the cost of considerable wastage of energy.

At this stage power, rather than matter, may limit the viable biomass. Assuming a power requirement of 100 Watt/kg biomass, this white dwarf star can support a biomass of 10¹³ kg, possibly in the form of 10¹¹ self-sufficient humans. The material for this biomass can be obtained from an asteroid or comet of 10¹⁵ kg with a radius of about 6 km. This is just one comet of the billions that exist at the present, some of which will survive the red giant phase of the Sun. Although some of the comets will be dispersed by passing stars, (Adams and Laughlin, 1999) a fraction will stay in the Solar System naturally. This may be also secured by technology by moving the comets closer to the Sun or by processing them into a Dyson Sphere (Dyson, 1979b and 1988).

This amount of population may have to be reduced by orders of magnitude if it captures only part of the stellar energy, or if they convert it with a low efficiency, or if there is a need for supporting industry or biomass. Using one percent of the stellar power, a population of one billion could each have access to 10 kW of power at living standards comparable to current industrial societies. The 5x10¹⁰ kg biomass of this population can be constructed using a small 10¹² kg asteroid or comet of 620 m radius.

If the low-temperature radiation of the star can be converted to habitable temperatures, lifestyles in this distant future can be Earth-like. From the point of view of the total integrated amount of life, the determining factors are the luminosity of the white dwarf Sun that can support 10¹³ kg biomass, possibly consisting of 10¹¹ self-sufficient humans during its lifetime of 10²⁰ years. This yields a time-integrated BIOTA_int of 10³³ kg-years possibly consisting of 10³¹  human-years of self-supporting humans or 10³⁰ human-years of humans each with a supporting biomass of 1,000 kg. By this scenario humans and a diverse biota can exist in our Solar System for an immensely long hundred million trillion years. The time-integrated biomass is a billion times more than the amount of life that has existed to date, and its time-span is more than ten billion times longer than of life on Earth to date.

These considerations assumed no wastage. However, the biomass constructed using the material resources would usually dissipate a fraction of its mass every year as wastage. On a long time-scale, even a minute rate of wastage can dissipate a very large amount of materials, and the resources must be able to cover this wastage. If the resources are not sufficient, either the rate of wastage per unit biomass, or the amount of biomass, or both, would have to be reduced.

Quantitatively, the relation is given by the equation M_biomassk_wastet = M_resourcec_{x(limiting), resource}/c_{x(limiting),biomass}. The terms are defined in Appendix 2.1. The concentration of the limiting resource in the cometary materials, c_{x(limiting), resource} is 60 g/kg for nitrogen. As an example, the 10²⁶ kg asteroid materials may need to sustain the biota and its wastage about the white dwarf Sun for 10²⁰ years. The sustainable rate of wastage k_waste x M_biomass is then 60,000 kg/year. If the 10¹⁵ Watt power output of the white dwarf Sun is all used, it can sustain 10¹³ kilogram of steady-state biomass, consisting of a population of 10¹¹ self-sufficient humans. In this case, the rate of waste must be as small as 6x10^-9 per year, i.e., a fraction wasted per year must be smaller than six billionth of the biomass. On the other hand, if a more realistic rate of waste of say 10^-4 y^-1 is assumed, then the steady-state M_biomass must be reduced to 6x10⁸ kg, consisting of a human population of at most ten million individuals. Compared with what can be sustained by the biomass and population by the energy of the star, the biomass and population must be reduced by a factor over ten thousand due to even this modest rate of wastage. Supplementary materials from outside the Solar System would be needed to maintain the much larger amount of life that the energy of the star can support. Similar considerations apply to other stars in the galaxy.

Resources and Population in Future Periods of Cosmology

The future evolution of the universe depends on dark matter and dark energy whose natures are largely unknown. Assuming a proton half-life of 10³⁷ years (Adams and Laughlin, 1999), the last proton of the 10⁴¹ kg of baryonic matter in the galaxy will decay after 1.6x10³⁹ years and of the 10⁵² kg matter in the universe, after a slightly longer 1.8x10³⁹ years. Other loss processes such as the incorporation of matter into black holes, or annihilation by dark matter, may also be possible. In comparison, the observable history since the Big Bang is one part in 10²⁹ of these time scales of 10³⁹ years. The current age of ordinary matter is approximately the same fraction of its future as the first 4.4 picoseconds (4.4x10^-12 seconds) in the past history of the universe. Evidently, the future behavior of matter, of the physical constants, and dark matter and dark energy cannot be predicted on the basis of this minute fraction of the future history.

If the universe expands constantly, biology will end but some abstract cognitive "life" may last forever (Dyson, 1979a). If the expansion rate accelerates as it appears to be happening now, even this will not be possible (Dyson, 2001). The present discussions concern the future of biological life only, quantified in the context of the cosmology of Adams and Laughlin (1999).

The expansion of life may depend on human endeavors. If life can survive in our Solar System around the white dwarf Sun for trillions of years, will humans still want to colonize other stars in the galaxy? Societies motivated by panbiotic ethics will strive to do so. In fact, life-centered ethics will help societies to survive, and societies that survive that long would evolve to value life. A purposeful civilization can then colonize the galaxy in a billion years, starting from this Solar System maybe, when impelled by the Sun at its red giant phase.

There will be several types of stars that can support and expand life. Their contributions to the total amount of life in the galaxy BIOTA_{int, galaxy} can be calculated using the equation: BIOTA_int, _galaxy = M_biomass, _star t_lifetime n_stars. Here M_{biomass, star} is the steady-state biomass about a given star that can be sustained by the material or energy resources, t_lifetime is the lifetime during which this biomass exists and n_star is the number of the given type of stars in the galaxy. Note that the product M_{biomass,  star} t_lifetime is the integrated biomass BIOTA_{int, star} about the given type of star. Of course, these terms may be uncertain by several orders of magnitude. In addition, each type of star will have a wide distribution of masses, luminosities, material resources and lifetimes, and therefore a wide distribution of the biomass and time-integrated biota that they can support. Future calculations must account for these distributions.

The first familiar environment will be red dwarf stars, with luminosities of 0.001 to 0.0001 times that of the Sun, that is, on the order of 10²³ Watts, which can support 10²¹ kg of biomass. This biomass can be constructed from planets, asteroids or comets if they can be found about red dwarf stars. If these resources are scarce, or if the rate of wastage is significant, then the materials may limit the biomass. Given enough materials, the sustainable integrated BIOTA_int with a life-time of 10¹³ years about a red dwarf is then on the order of 10³⁴ kg-years, and 10¹² red dwarfs in the galaxy will allow a BIOTA_int of 10⁴⁶ kg-years.

Brown dwarfs may also accommodate life. These small stars are lighter than 0.08 M_Sun but somewhat heavier than the gas planets. They radiate heat slowly due to gravitational contraction with a typical luminosity of 10²⁰ Watts, which can support 10¹⁸ kg of biomass for the 10¹⁰ year lifetime of these stars. This contributes 10²⁸ kg-years of potential integrated biomass per star, and the 10¹¹ such stars in the galaxy contribute 10³⁹ kg-years of integrated biomass.

In the long term, collisions between brown dwarfs will give rise to the last red dwarf stars, possibly with habitable planets (Adams and Laughlin, 1999). The total power output of these last stars in the galaxy will be similar to that of a single star like the Sun, on the order of 10²⁶ Watts (Adams and Laughlin, 1999) supporting a total of 10²⁴ kg biomass. With lifetimes of 10¹⁴ years they can contribute 10³⁸ kg-years to the integrated biomass in the galaxy. The amount is small compared with other ecosystems, but the significance is that this mode of star formation will produce liveable environments for a long time.

The longest lasting stars in the galaxy will be the white dwarfs. Most of the stars that have ever formed in the galaxy will end up at this stage, yielding on the order of a trillion, 10¹² such stars (Adams and Laughlin, 1999). As discussed above for our Sun, each can yield a BIOTA_int of 10³³ kg-years giving a BIOTA_int of 10⁴⁵ kg-years in the galaxy.

The estimates of biomass for each ecosystem in the galaxy can be extended to the universe by multiplying it by the estimated 10¹¹ galaxies. Unless there is local life in these galaxies, they must be reached by colonizing life-forms while they are within the accessible event horizon. If we reach them, biology and human life may exist in the galaxies after they become in effect separated universes. Life may have originated in these galaxies, or life must reach them before they become unreachable because of the expansion of the universe. Our family of life, even branches of humankind, may then exist in billions of universes that will be permanently separated from each other, beyond communication.

These calculations concern upper limits of biomass and populations, i.e., the carrying capacities of the ecosystems as determined by resources of mass or energy. In fact, the populations that will be realized may be limited by the mechanism and rate of expansion of life. Adams and Laughlin (1999) suggested that a natural "random walk" mechanism to populate the galaxy would take trillions of years, while purposeful colonization may succeed in a billion years. Until the difficulties of human interstellar travel are overcome (Mauldin, 1992), life may be spread through directed panspermia, even by current-level technology. The cometary materials in our Solar System are sufficient to seed with microorganisms all the new planetary systems that will form in the galaxy during the next five billion years of the Sun (Mautner and Matloff, 1977 and 1979, Mautner, 1997). The maximum rate of growth of biota in the galaxy, i.e., the biotic potential, is therefore likely to depend on human will and technology rather than on natural limitations.

The above calculations illustrate the immense amounts of potential future life and the considerations that may be applied to quantify these amounts. These estimates are uncertain by orders of magnitude and will be re-evaluated as cosmology advances.

The Ultimate Amounts of Life in the Universe

Finally, we may ask “What is the maximum amount of biological life that the universe could support?” In terms of material resources, the maximum biomass is achieved if all ordinary baryonic matter is converted to elements in their biological proportions, and these elements are then incorporated into biomass. The amount of baryonic matter may be estimated from the mass of the Sun, 10³⁰ kg, multiplied by the 10¹¹ stars and 10¹¹ galaxies, yielding 10⁵² kg. A more sophisticated calculation based on the volume of the universe in an event horizon of 15 billion light-years and the estimated density of baryonic matter of 4.1x10^-28 kg m^-3 yielded a similar result of 5.9x10⁵¹ kg (Wiltshire, 2002).

To maximize Life, all of the 10⁵² kg baryonic matter would be converted to biomass. However, this would not leave any sources of energy, except dark matter, gravitational energy and dark energy and background radiation, which may be impractical to utilize. As a source of energy, a portion of the biomass would have to be converted to energy at a rate that provides the required power for the remaining biomass. The maximum energy may be produced according to e = mc² by the relativistic conversion of mass. The calculations of the rate of use of matter in this manner are described in Appendix 2.2. To supply a power of 100 Watts per kg biomass, a fraction of 3.5x10^-8 of the mass must be used per year. The remaining biomass at time t can be calculated from equation (A5) in Appendix 2.2 (substituting k_waste by k_use = 3.5x10^-8 year^-1). At this rate, all of the 10⁵² kg matter of the universe will be reduced to the 50 kg biomass of the last human after 3.3 billion years, and to the last 10^-15 kg microorganism after 4.4 billion years. The total integrated BIOTA_int will be 3x10⁵⁹ kg-years, possibly in the form of 6x10⁵⁷ human-years. Similarly, if only the 10⁴¹ kg of baryonic matter in the galaxy is available, it will be reduced to the last human after 2.6 billion years and to the last microorganism after 3.7 billion years. The total integrated BIOTA_int will be 3x10⁴⁸ kg-years, possibly in the form of 6x10⁴⁶ human-years.   

Converting all the baryonic mass to biomass at the fastest technological rate would achieve the cosmic biotic potential, i.e., accomplish the maximum growth rate of life in he universe. Sustaining the biomass by converting a fraction to energy would in turn achieve the maximum time-integrated biomass in the universe. However, matter for life will have been exhausted and life would become extinct after a few billions of years. This time is much shorter than the time allowed for by proton decay.

As discussed above for the Solar System, the same time-integrated BIOTA_int can be achieved with the same resources over a much longer time-span by constructing the biomass at a slower rate and maintaining a smaller steady-state population. For example, assume that we wish to use the matter in the galaxy at a rate that would allow life to exist for the 10³⁷ years allowed by proton decay. At the required conversion rate of mass to supply energy, this would allow a steady-state biomass of 3x10¹¹ kg, possibly in the form of 6x10⁹ humans, yielding over 10³⁷ years 3x10⁴⁸ kg-year of time-integrated biomass, possibly as 6x10⁴⁶ human-years,  in the galaxy. While life will have existed much longer, the time-integrated biomass remains the same as in the previous scenario of rapid construction and use. Extrapolating to the universe, life could similarly exist for 10³⁷ years with a steady-state biomass of 3x10²² kg possibly in the form of 6x10²⁰ humans. This would yield 3x10⁵⁹ kg-years of time-integrated biomass, possibly as 6x10⁵⁷ human-years, lasting during the 10³⁷ years that baryonic matter exists in the universe.

These scenarios describe the maximum amount of biological life that appears possible according to current cosmology. Even if a small fraction of this amount is realized, it will be immensely greater than the amount of life that has existed to the present.

 Estimated resources, biomass and time-integrated biomass (BIOTA_int) supported by the principal resources of future periods of cosmology.

Footnotes

1. Per solar system.
2. Biomass obtained using extractable elements in carbonaceous chondrite-like asteroids or comets, based on N and P as limiting nutrients.
3. Biomass obtained using total elemental contents in carboanceous chondrite-like asteroids or comets, based on N and P as limiting nutrients. It is assumed that by the red giant phase of the Sun the total contents of cometary materials will be available technologically. The numbers are order-of-magnitude estimates, as amounts of resource materials and power are not known accurately.
4. Biomass based on power supply of 100 Watts/kg as the liming factor.   
5. Per galaxy.
6. Total in the universe.

Conclusions

In summary, carbonaceous chondrite materials in asteroids and comets are the most likely resources in the Solar System.  The limiting nutrients in these resources such as nitrogen or phosphorus may determine the total integrated biomass that can exist during the next five billion year main sequence phase of the Sun. On these time scales wastage is also critical. Minimizing the rate of wastage to 0.01% of the biomass per year would allow a permanent human population of several billions during this period. Further projections are speculative, but biological life may survive the Red Giant phase of the Sun and may continue under the White Dwarf Sun for 10²⁰ years. Ultimately, the span of biological life may be limited by the sources of energy and by the proton decay, but these time spans may reach to 10³⁷ years.

Current cosmology requires us to re-examine our ethics. In particular, it must be considered that biological life, or even other abstract "life", has finite duration. However, the physics are uncertain, and more solid predictions may require trillions of years of observation. If life is indeed finite, a measure is needed to quantify its amount. Time-integrated biomass was used here. The calculations showed that the potential amounts of life in the cosmological future are immensely greater than life that has existed to date. Our family of organic life can survive in great numbers for immensely long times.

A Life-centered ethics will be necessary to reach that future. With such motivation we can populate space and start new chains of evolution throughout the galaxy. Astroecology and cosmology suggest that these programs can lead to an immense expanse of life that can give human existence cosmic consequences. Our remote descendants may explore if Nature can be transformed in their favor so that Life can exist forever.

Astroecology Calculations

Resources and Biomass

Living organisms take up nutrient elements from resource materials and incorporate them in biomass. Various types of resource materials contain various concentrations c_x,resource of each nutrient element x, reported in Table A2.2 in units of g/kg. Similarly, c_x,biomass (g/kg) is the concentration of element x in a given type of biomass as summarized in Table A2.3. Equation (A1) gives the amount of biomass, m_x,biomass (kg) that could be constructed from an amount m_resource (kg) of resource material if element x was the limiting factor and the other components of elements were available without limitation.

                        m_x,biomass = m_resource c_x,resource / c_x,biomass                 (A1)

            Table A2.4 lists the amounts of biomass (kg) that can be constructed from each nutrient in 1 kg of the resource materials. Note that more than 1 kg of biomass could be constructed from 1 kg of resource materials as based on element x, for example, if a rare nutrient x is over-abundant in the resource materials.

Rates of formation, steady-state amounts, and total time-integrated biomass.

We consider ecosystems where biomass M_biomass (kg) is formed at a constant rate dM_biomass/dt = k_formation (kg y^-1) and is used up or wasted at a rate dM_biomass/dt = -k_waste M_biomass (kg y^-1). In astroecology, the formation may represent conversion of space resources to biomass while usage or waste may occur through leakage to space, or by the conversion of a fraction of the biomass to energy to provide power for the remaining biomass.

Note that the formation rate is zero order and the rate of waste is first order in M_biomass. Equation (A2) gives the rate of change of the biomass.

                        dM_biomass/dt = k_formation - k_wasteM_biomass                  (A2)

Note that k_waste in units of y^-1 represents the fraction of M_biomass that is wasted per year. At steady state the rate of change of the biomass is zero e.g., dM_biomass/dt = 0, and equation (A3) gives the steady-state biomass M_{biomass, equlibrium} (kg).

            M_{biomass, steady-state} (kg) = k_formation (kg y^­-1) / k_waste (y^-1)        (A3)

Next we calculate the time-integrated BIOTA_int (Biomass Integrated Over Times Available) that exists in the ecosystem during a finite time period. In equation (A4) we integrate M_{biomass, t} i.e., the biomass at any time, from the starting time t_o of the ecosystem to the end time t_f of life in the ecosystem.

                                                      t_f

BIOTA_int = ∫ M_{biomass, t} dt                                              (A4)    

                                                                      t_o

After the formation of a given amount of biomass M_{biomass, o} has been completed, it may be used up or wasted at the rate of -k_waste, i.e., the remaining amount of this unit of biomass decreases according to equation (A2) with k_formation = 0. The solution in equation (A5) gives the instantaneous amount that remains of this unit of biomass after time t.

M_{biomass, t} = M_{biomass, o} exp(-k_waste t)                     (A5)                

By integrating equation (A5), we obtain the total integrated amount of this amount of biomass that will have existed from its formation to infinity. 

BIOTA_int = M_{biomass, o}/k_waste                               (A6)

Note that equation (A6) applies to each unit of biomass that decays at the rate of -k_wasteM_biomass regardless of when it was formed. Therefore, the total integrated biomass of the ecosystem depends only on the total amount of biomass created and on the decay rate, but not on the rate of formation. Equation (A7) gives the total time-integrated biomass BIOTA_{int,ecosystem} of the entire ecosystem. If the total amount of biomass created during the lifetime of the ecosystem is M_ecosystem then

BIOTA_{int, ecosystem} (kg-y) = M_ecosystem (kg) / k_waste (y^-1 )     (A7)

            Note that if waste is reduced to zero and no mass is lost from the biosystem then k_waste = 0 and the integrated BIOTA_int is infinite for any finite amount of biomass. At the extreme, a single bacterium living forever would give an infinite amount of integrated biomass.

            If all the mass M_resource of the resource materials is converted to the maximum biomass that is allowed by the limiting nutrient according to Equation (A1), then equation (A8) gives the total time-integrated integrated BIOTA_{int, ecosystem} of the ecosystem. 

BIOTA_{int, ecosystem} = (M_x,resource c_x,resource/c_x,biomass)/k_waste    (A8)

            An interesting case occurs if a fraction of the biomass is used to provide energy for the remaining biomass. Assume that the power requirement is P_biomass (J s^-1 kg^-1) and the energy yield is E_{yield, biomass} (J kg^-1) per unit (kg) biomass converted to energy. If the biomass is converted to energy at the rate required to provide the needed power for the remaining biomass, then

(-dM_biomass/dt) (kg s^­-1) E_{yield, biomass} (J kg^-1) = 

P_biomass (J s^-1 kg^-1) M_biomass (kg)                          (A9) 

            This is similar to equation (A2) with a formation rate of zero and with k_{waste =} P_biomass/E_{yield, biomass}.

The remaining biomass after time t is given according to equation (A5) as

M_{biomass, t} = M_{biomass, o} exp (-(P_biomass/E_{yield, biomass}) t)                    (A10)

            The maximum energy can be obtained from a unit of mass by conversion to energy according to the relativistic relation E = mc². In this case E_{yield, biomass} = c² and assuming a power need of P_biomass = 100 Watt/kg, the decay rate of the biomass is

k_use = 100 (J s^-1 kg^-1) / (3 x 10⁸)² (m² s^-2) =

1.11x10^-15 s^-1 = 3.5x10^-8 y^-1                    (A11)

For a simple estimate of the amount of baryonic matter in the universe, the 10³⁰ kg mass of the Sun may be multiplied by the 10¹¹ stars and 10¹¹ galaxies, yielding 10⁵² kg of baryonic matter. A more sophisticated calculation that was based on the volume of the universe (event horizon with a radius of 1.5x10²⁶ m and volume of 1.4x10⁷⁹ m³) and the density of baryonic matter, 4.1x10^-28 kg m^-3 lead to a similar result of 5.9x10⁵¹ kg (Wiltshire, 2002).

If all the baryonic matter in the universe were converted to elements according to their proportions in biomass, then this process would yield 10⁵² kg of biomass. If a fraction of this biomass was converted to energy at the rate shown in equation (A11), then the initial rate of mass loss would be 3.5x10⁴⁴ kg y^­-1. There would be enough biomass left for one 50 kg human being after 3.3x10⁹ years, and for a single bacterium of 10^-15 kg after 4.4x10⁹ years. The total time-integrated life will have been 2.8x10⁵⁹ kg-y.

It is unlikely of course even in principle that all the matter in the universe can be brought together in one biosphere, since the galaxies are receding beyond their mutual event horizons. By analogous considerations, the duration of life using the 10⁴¹ kg matter in each galaxy is 2.6x10⁹ years to the last human and 3.7x10⁹ years to the last microbe. The total integrated life is 2.8x10⁴⁸ kg-y per galaxy, which yields  2.8x10⁵⁹ kg-y in all the galaxies. This amounts to 5.6x10⁵⁷ human-years, or 5.6x10⁵⁵ humans who will have each lived for 100 years. Although these numbers are not realistic and have large uncertainties, they illustrate the upper limits of biological and human life in the universe.

Energy Flux, Temperatures and Habitable Zones about Stars

The luminosity of a star is equal to its power output, i.e., its energy output per unit time. The power output is related to the surface temperature according to

                        L (J s^-1) = 4пr_S²σT⁴                               (A12)

Here L is the luminosity, 4pr_S² is the surface area and T (^oK) is the surface temperature of the star, and σ is the Stefan-Boltzmann constant, 5.67x10^-8 J m² s^-1 K^-4. The radius of the Sun is 6.96x10⁸ m and its luminosity is 3.9x10²⁶ J s^-1.

            A spherical object with a radius r, at a distance R from the Sun, absorbs the solar flux intercepted by its projected area, at the rate

w_abs = (L/4πR²) (πr²) (1-a)                                 (A13)

Here the terms in the first parentheses give the solar energy flux at the distance R, the second parentheses the projected area, and the third parentheses account for the albedo, that is, reflection of part of the radiation.  

            The object also emits radiation depending on its radius and temperature according to equation (A12). At steady-state the absorbed and emitted radiation are equal and equations (A12) and (A13) can be combined to give the steady-state temperature as equation (A14).

                        T⁴ = L(1-a)/16πR²σ                                           (A14)

            The steady-state temperature defines the habitable zone. It may be considered for example as the zone in which the steady-state temperature allows the existence of liquid water at 1 atmosphere total pressure, that is, 273 to 373 K (i.e., 0 to 100 C). More generally, we could define a narrower "comfort zone" say at 298 K (i.e., 25 C) or a wider "survivable zone" where life can exist with reasonable technology, for example, where the temperature ranges between 200 to 500 K. 

            We can use equation (A14) to calculate the habitable zone about the star as its luminosity changes. For example, it is estimated that in the red giant phase the luminosity of the Sun will increase to 2,000 times its present value. Then the habitable zone, for objects with an albedo of 0.3, will be between 21 and 39 au, and the comfort zone at 298 K at 33 au. At a later stage, the luminosity of the Sun may increase to 6000 times its current value. The habitable zone will then be from 36 to 68 au and the comfort zone will be centered about 57 au. Since the Sun will lose some mass and the planets will move out to larger orbits, Neptune, Pluto and the inner Kuiper Belt objects may move into or near the habitable zone at that time.   

Pluto and the other Kuiper Belt objects are cometary nuclei that contain various ices. We may assume that objects that remain below 150 K will not evaporate, except for losing some highly volatile substances. Even at the hottest stage of the Sun, objects further than 225 au will remain at these low temperatures. This applies to the outer Kuiper belt which may extend to 1,000 au, and of course to the Oort cloud comets at 40,000 au, which will be only at 11 K even when the Sun is the most luminous. The preserved matter in the outer Kuiper Belt and the Oort Cloud may contain on the order of 10²⁴ - 10²⁵ kg of materials including organics and water that may be used as biological resources.

Note that travel to the 100 au comfort zone when the Sun is most luminous, at a reasonable speed of 10^-4 c lasts only 16 years, and even to 1000 au to collect Kuiper Belt resources only 160 years. These resources can be accessed by humans with contemporary or slightly extended life-spans. In comparison, travel to the Oort clout at 40,000 au will last over 6,000 years and interstellar travel to nearby habitable stars will last from tens of thousands to millions of years. Such travel will require major changes in technology and biology. It is comforting therefore that humans like ourselves may survive in this Solar System through its hottest days, and that life can continue in our Solar System for inconceivably long hundreds of trillions of years during which the white dwarf Sun can sustain life.

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