iBook The Future of the Universe - 5. Rejuvenating the Sun

5. Rejuvenating the Sun 

In Chapter 3 the physical processes underlying the workings of a Sun-like star were described. In this chapter we will examine the ways in which the properties of a star might, at least in principle, be manipulated by our distant descendants. Specifically, our task is to see how the Sun might be ‘engineered’ or ‘rejuvenated’ to enable the continued survival of life on the innermost planets, on timescales greater than the canonical main-sequence lifetime [T > TMS (canonical)]. In the case of Venus and Mars, of course, this clearly means future human life on terraformed worlds. As already stated, the task of the would-be asteroengineer is to find ways to stop the Sun from becoming over-luminous, and from becoming a bloated red-giant – the dire consequences of these effects for the Solar System having been discussed in the last chapter. It turns out, fortuitously for humankind, that these goals are compatible; by stopping the red-giant Sun from coming about, the long-term temperature stability of the inner planets is also maintained. Perhaps it should be reiterated at this stage that we are not describing in this book exactly how the mechanical part of star engineering can be done. We do not know, for example, what kinds of materials should be used or how to construct the various machines and devices that will be described in this chapter. What we will outline, however, is how the future properties of the Sun might be controlled in principle. The Engineering Options As highlighted in Chapter 4, the most important problem that the future star engineer will need to address is that of the Sun’s increasing luminosity. Its increase in radius is not so great an issue if we are only concerned with the survival of Planet Earth, but it seems an incredible waste of resources to simply let Mercury 147 148 Rejuvenating the Sun and Avoiding Other Global Catastrophes and Venus be consumed by an expanding Sun. Equation (4.1) contains the key terms of interest and, indeed, it indicates that for a fixed planetary distance d, the surface temperature of the planet increases as the Sun’s luminosity to the one-quarter power—that is as L1/4. If all of the other terms on the right-hand side of Equation (4.1) remain the same, then the surface temperature of any given planet increases by about one degree for every 1 percent increase in the Sun’s luminosity. So, to stop Earth from overheating, the star engineer must control the growth of the Sun’s luminosity. Indeed, the aim will be to keep the Sun at or at least near its present energy output per unit time. In fact, a slightly less luminous Sun might be desirable. This latter dictate builds upon the suggestion by Professor James Lovelock that the recent glacial – interglacial cycling that has dominated the Pleistocene era is a Gaian response to the enhanced warming of Earth in recent times.1 The Sun’s ideal luminosity was achieved, according to Lovelock, some 2 billion years ago, when it was 15 percent less luminous than now. How then might the star engineer proceed? The double goal of eliminating the red-giant phase and reducing the Sun’s luminosity—the basic act of rejuvenation—can be achieved by manipulating both internal and external quantities. By external quantities we specifically mean the mass of the Sun, and by internal we mean the radial variation in its composition. No one process of manipulation is going to achieve both of the stated goals, so a combination of alteration mechanisms will be required. Mixing and Mass Loss In this section we will build upon the results leading to Equation (3.12). Specifically the mass-luminosity relationship indicates that if the Sun is to have the same luminosity at the beginning and the end of its main-sequence phase, then its mass at the end of the main-sequence must be reduced to2 ∼ 0.3 M In other words, the Sun must be slimmed down by some 0.7 Mworth of material. The example considered above assumes that the Sun has a homogeneous composition. Detailed numerical models, however, Rejuvenating the Sun 149 have shown that even if a star has an inhomogeneous composition (where the envelope, for example, is more hydrogen-rich than the core), the evolution with mass-loss is always at a lower luminosity. Figure 5.1 illustrates, in a schematic way, the effects of mixing and mass-loss on the evolution of a star.3 It can be seen from the figure that the effect of inducing greater and greater amounts of additional mixing within the interior of a star results in the red-giant phase being killed off. Rather than evolving into a low temperature, large red-giant at core hydrogen exhaustion, a fully mixed star evolves into a luminous, slightly larger, and higher temperature star. To the star engineer this result illustrates how the bloated red-giant stage of the Sun can be avoided and, accordingly, methods of mixing the Sun’s interior will have to be developed. The evolution of a fully mixed star with massloss is again toward higher temperatures, but now the mass-loss ⇐ Temperature Main sequence Fully mixed No mass loss Canonical evolution No mass loss Increasing mass loss Increasing mass loss Increased mixing ⇑ No mass loss Luminosity Figure 5.1. Schematic diagram showing the effects of mass-loss and mixing on the evolution of a star.3 The standard non-mixed evolutionary tracks are shown as dashed lines. The chemically homogenous evolutionary tracks are shown as solid lines. The effect of partially mixing a star is shown by the dotted lines. The shaded region indicates the effect of mixing without mass-loss. 150 Rejuvenating the Sun and Avoiding Other Global Catastrophes results in lower luminosities being achieved—the greater the massloss, the lower the luminosity for any given composition. If the mass-loss is very high, the evolution can proceed to values lower than the initial main-sequence luminosity. A non-homogeneous star evolving with mass-loss is also, for a given composition, less luminous than the non-homogeneous zero mass-loss model. The evolution is still towards lower surface temperatures, however, and unless extreme amounts of mass are removed from the star, the red-giant phase will still occur. To sum up so far, for the Sun to avoid its bloated red-giant phase, and for it to evolve at near constant luminosity, both mass-loss and the (near) complete internal mixing of its chemical elements must be engineered. Adding to the Pressure The extent of a star’s hydrogen-burning phase (its main-sequence lifetime) is expressed, constants aside, by Equation (2.1) as TMS ∼ M/L. For a star of fixed mass M this relationship indicates that the hydrogen-burning phase might be extended if the star can be made to operate at a lower luminosity L. Such a possibility exists, provided that the would-be star engineer can find a way of introducing some non-thermal pressure support to help the star remain in hydrostatic equilibrium at a lower temperature. In this manner the pressure P(r) at any point within a star is composed of two terms: the gas pressure Pgas and a non-thermal pressure, PNT. Two examples of non-thermal pressure support relate to strong magnetic fields and rapid internal rotation. If the pressure is written as P(r) = Pgas + PNT = Pgas(1 + ), where = PNT/Pgas is the ratio of the non-thermal to the thermal pressure support, then Equation (3.5) for the central pressure can be re-derived, and accordingly4 TC = [G mH/3 k] μ(1 – ) M/R. When = 0 there is no non-thermal pressure support, and we recover Equation (3.5). As increases above zero, however, the central temperature TC required to achieve hydrostatic equilibrium is reduced. The reduced central temperature that results from the introduction of an additional non-thermal pressure term dictates that the PP chain will run less efficiently [recall Equation (3.11)] Rejuvenating the Sun 151 and, consequently, the luminosity will be reduced. Indeed, the mass-luminosity relationship for Sun-like stars [Equation (3.12)] becomes L = LKR[μ(1 – )]75 M5/(1 + X). Now we recover the result that as increases from zero so, for a given composition and fixed stellar mass, the luminosity is reduced. If we go back to the expression for the main-sequence lifetime, TMS ∼M/L, then the effect of introducing additional non-thermal pressure support is to increase the main-sequence lifetime to Tntps = TMS (1 – ) −75. Table 5.1 indicates the effect of introducing increasing amounts of non-thermal pressure support. The greater the value of , the longer the main-sequence lifetime and the lower the luminosity of the star at core hydrogen exhaustion. Indeed, as Table 5.1 indicates, if the non-thermal pressure support in the Sun could be increased to a value of order 10 percent, then its main-sequence lifetime would be increased by a factor of two, to of order 20 billion years, and its luminosity at core hydrogen exhaustion would be reduced by a factor of about one-half compared to its canonical (and fully mixed) evolutionary value at core hydrogen exhaustion. To drive the central temperature of the Sun down by engineering additional non-thermal pressure support would be far from simple—at least to begin with. The Sun’s magnetic field is currently generated within its convective outer envelope, and there is no straightforward way to sustain a central magnetic field. By inducing mass-loss, however, (as indicated in Figure 3.6) the rejuvenated Sun will develop a deeper and deeper outer convection zone as its mass is physically reduced. Once the mass is below about 0.5 M then the modified Sun will be nearly fully convective, and the enhanced magnetic dynamo action that will then operate should Table 5.1. The effect of additional non-thermal pressure support upon the main-sequence lifetime. The first column indicates the value of . Columns 2 and 3 show the resultant increase in the main-sequence lifetime and the reduced luminosity (when X = 0) for non-zero values of . Tntps/TMS L(X=0, )/L(X=0, = 0) 0.01 11 091 0.05 15 067 0.1 22 045 0.2 53 019 0.4 461 002 0.5 1810 0005 152 Rejuvenating the Sun and Avoiding Other Global Catastrophes result in the generation of a significant non-thermal pressure effect at the center. If this late-stage rejuvenation process can be made efficient, then the mass-loss rate could be modified downward, since the non-thermal pressure support provided by a strong central magnetic field will cause a reduction in the Sun’s luminosity. Although the non-thermal pressure support due to magnetic fields might well become important during the later stages of the Sun’s rejuvenation (once its mass is reduced below about 0.5 M), rotation-related non-thermal pressure support might conceivably be induced during the early rejuvenation stages. In this case the Sun’s internal gravity is counteracted by the centrifugal repulsion that comes about because of the rotation. The faster the Sun can be made to rotate the greater the rotational non-thermal pressure support. In addition, detailed numerical modeling indicates that enhanced rotation within a star leads to more extensive mixing of the chemical elements and in the enhancement of the overall magnetic field. Both of these results are desirable star-engineering byproducts. As we will see in more detail below, David Criswell has described a method by which a star might be spun-up and ‘mined’ at the same time. The Opacity Effect A second method by which a star’s luminosity might conceivably be reduced, thereby enhancing its hydrogen-burning lifetime, is through the manipulation of the opacity of stellar material. As described in Chapter 3, the opacity is a measure of how much the stellar material hinders the passage of radiation through it. The greater the opacity of a region, the longer it takes the radiation to pass through it. The effect of increasing the opacity is to reduce the mean-free path length l = 1/( ) between photon interactions. This, in turn, results in a longer photon diffusion time TPD [as given by Equation (3.6)]. Examining Equations (3.7) and (3.9) further indicates that an increase in the opacity results in a lower surface temperature and a smaller luminosity. In other words, the greater the opacity of stellar material, the cooler and less luminous a star will become. The lower temperature and luminosity, however, will result in the star becoming larger, in accordance with the Rejuvenating the Sun 153 Stefan-Boltzmann law [as given by Equation (3.8)]. In a technical, but very readable, paper on the order-of-magnitude theory of stellar structure,5 Professor George Greenstein of Amherst College shows that if the opacity is increased by a factor of 100 over that provided by electron scattering, then the luminosity will be reduced by a factor of 100 and the surface temperature will be reduced by a factor of about 10. A significant fraction of the stellar opacity within Sun-like stars is due to the heavy elements (measured by the Z abundance; see Table 3.1). Simply adding additional heavy element-rich material to the Sun will not actually help in reducing its luminosity, since this process only adds more mass to the Sun, and this will offset the required effect as indicated by the massluminosity Equation (3.12). One might envision a dialysis-like process, however, whereby solar helium is extracted and then preferentially replaced with, say, iron or some such similar heavy element. This, however, would be an exceptionally complicated process, and there is no ready-at-hand supply of heavy element material with which to effectively seed the Sun. The Tools of the Trade A summary of the basic rejuvenation processes described above are presented in Table 5.2. The processes apply not just to the Sun, but to all stars that are likely to harbor habitable planets and possibly advanced life forms. There are a few additional engineering options that will be introduced below, but for the moment it seems6 that the best way to begin the process of rejuvenating the Sun, and thereby saving Earth from complete devastation, is to engineer additional mixing within its interior and enhance its mass-loss rate. The non-thermal pressure support and opacity alteration methods may still be employed at a later time, but they might be more advanced processes (both physically and temporally) in the rejuvenation sequence. In short, the solar rejuvenation process will initially proceed by manipulating the physical body of the Sun. Indeed, just like a human being, the Sun can be rejuvenated by slimming down and consuming its food (in the form of hydrogen, of course) more efficiently. 154 Rejuvenating the Sun and Avoiding Other Global Catastrophes Table 5.2. Summary of the principal processes available for rejuvenating a star. Process Reason [key equations] Response Mass reduction Mass-luminosity relationship [(3.12)] Luminosity reduction Non-thermal pressure support Reduced thermal pressure requirement at center results in lower central temperature [(3.5), (3.7), (3.9), (3.11)] Reduction in central temperature leading to a reduced luminosity and lower surface temperature Opacity increase Reduced mean-free path length (l) [(3.6), (3.7) and (3.9)] Reduction in luminosity and surface temperature Mixing of elements Avoids a core–envelope composition discontinuity [see Figure 5.1] Kills off red-giant stage expansion A Homogeneous Star Model In Chapter 3 the mass-luminosity relationship for homogeneous Sun-like stars was described [see Equation (3.12)]. Accordingly, the luminosity L is related to the mass M and the chemical compositions as L = L0 μ 75 (1 + X) −1 M 5, where L0 = L(X = X0) is a constant, μ is the mean molecular weight [see Equation (3.4)] and 0 ≤ X ≤ X0 is the hydrogen mass fraction of the stellar gas. In Appendix A it is explained that when the mass fraction of the chemical elements other than hydrogen (X) and helium (Y) are small, then the mass-luminosity relationship can be written as L(X)≈ L0 (1 + X) −16 M5. This modified formula7 expresses the luminosity of a fully-mixed star of mass M solely in terms of the hydrogen mass fraction X. As a star ages, however, and hydrogen is consumed via the proton-proton chain of fusion reactions, the value of X will decrease from its initial X0 = 0.7 to zero. For a constant mass star, therefore, L(X = 0)/L(X = X0) = 1/(1 +X0) −16 ≈ 4900. In other words, for a fully mixed star, the luminosity is expected to increase by a factor of several thousand during its main-sequence (that is hydrogen-burning) lifetime. Detailed numerical models confirm Rejuvenating the Sun 155 this brightness increases to about the correct order of magnitude8. Simply fully mixing a star will not produce the rejuvenation effect being sought, and while the effect will kill off the giant phase it will not stop the Sun from becoming over-luminous. Introducing Mass-Loss Mass-loss, literally the reduction in the mass of a star, is an observed phenomenon. The Sun, for example, currently loses mass via the so-called solar wind, while other stars are observed to spew so much material into space that they become surrounded by glowing nebulae. It is the solar wind that is responsible for producing auroral displays in Earth’s upper atmosphere and for occasionally interrupting the workings of communications spacecraft. The solar wind emanates from the Sun’s outer corona and careens past Earth with a speed that varies between 200 and 800 km/s. The mass-loss rate for the Sun amounts to some 1.5 x 10−14 M/yr, equivalent to about a billion kilograms of material being ejected into space per second. The current annual mass-loss rate from the Sun is minuscule compared to its total mass, and even if it continued to lose mass at its current rate for the rest of its main-sequence lifetime, then only 10−4M of material would be lost into space. This, however, is equivalent to the mass of about 53 Earths. Observations indicate that the mass-loss rate varies in proportion to the luminosity of a star, and it is useful to express the mass-loss rate via the expression M/ t = N (L/c2), where M is the amount of mass lost by the star in the time interval t. N is a numerical parameter that can vary from zero (indicating no mass-loss) to a value as high as several hundred, L is the luminosity, and c is the speed of light. Using this expression for the mass-loss rate it can be shown (see Appendix A) that the mass of star decreases exponentially as m = exp[ k (x – 1) ] , where m=M(X)/M(X=X0), x = (1 + X)/(1 + X0) and k is a mass-loss rate dependent term defined in Equation (A.7). We now have the result that the mass of a star decreases as X decreases from X0 to zero, and that the larger the value of the mass-loss parameter N, the more rapidly does the mass decrease with X. Introducing the notation l = L (X)/L(X = X0), the luminosity of a chemically homogeneous, 156 Rejuvenating the Sun and Avoiding Other Global Catastrophes mass-losing star can be expressed as l = x−16 m5, and from this formulism we see that a mass-losing star is always going to be under-luminous when compared to a star of similar composition, but not losing mass.9 It is this under-luminosity condition that the future star engineer will want to exploit. Indeed, with respect to rejuvenating the Sun and extending the habitable lifetime of Earth, inducing additional mass-loss from the Sun is going to be crucial. The Fate of the Ejected Material Earth’s average global temperature is described by Equation (4.1), and Tsurface will vary according to how the L/d2 term changes with time. (Let’s assume for the sake of argument that Earth’s albedo and emissivity remain constant.) If the orbital distance d remains fixed, then Tsurface remains constant, provided the Sun’s luminosity L is constant. If, on the other hand, the orbital distance d increases with time, then Tsurface will only remain constant if the Sun’s luminosity L increases in order to keep the L/d2 term fixed. Since the solar rejuvenation process invokes mass-loss to control the Sun’s luminosity, two extreme scenarios can be explored, with the ultimate location of the mass removed from the Sun being the controlling factor. Accordingly, 1. If the material removed from the Sun is fully contained within the region interior to Earth’s orbit, then Earth’s orbital radius will remain fixed, and Tsurface will remain constant provided the Sun’s luminosity L is kept constant. 2. If the material ejected from the Sun is lost into interstellar space (or at least escapes to the outer Solar System), then Earth’s orbital radius will increase in accordance with the conservation of angular momentum, and Tsurface remains constant provided the ratio L/d2 = constant is maintained. In this scenario the Sun’s luminosity is allowed to increase with time. In Scenario 2 it is the conservation of angular momentum that will determine the Sun-Earth separation d(t). The increase in Earth’s orbital radius with decreasing solar mass is expressed as d(t) = d(0) exp[-k(x – 1)], where the constants k and x are described Rejuvenating the Sun 157 in Appendix A, and d(0) is Earth’s orbital radius at the onset of the rejuvenation process. For a fully mixed star to evolve under the constant luminosity condition (Scenario 1), the mass-loss rate must vary according to the parameter N1 (given by Equation A.9). For a fully mixed star to evolve with L/d2 = constant (scenario 2) the mass-loss rate must vary according to the parameter N2 (given by Equation A.10). Scenario 2, as stated before, seems to be an unnecessarily wasteful process. Why not use all or most of the material being removed from the Sun to build or power something else? If some of the material being driven from the rejuvenated Sun can be channeled to Jupiter, for example, then a low-mass synthetic star might conceivably be generated. The minimum mass for the initiation of hydrogen conversion as a bona fide star is about 0.1 M (recall Figure 3.15). If the mass of Jupiter was, therefore, increased by a factor of about 100, it could begin to generate sustained energy to power industrial plants on the Jovian moons and within the main-belt asteroid region. This being said, by generating a lowmass binary companion to the Sun, one would have to be exceptionally careful not to catastrophically alter the orbital dynamics of the planets and asteroids within the inner Solar System. The devil, as always, is in the details. An Outline Scenario for Rejuvenating the Sun In this section we will consider the requirements for rejuvenating the Sun under the two mass-loss scenarios described above. For the sake of convenience let’s assume that the Sun has been successfully homogenized and that the initial hydrogen mass fraction is X0 = 0.7. Table 5.3 shows the variation in the mass-loss rate as well as the rejuvenated Sun’s mass and surface temperature,10 as a function of X for Scenario 1, where it is just the luminosity that remains constant. Table 5.4 indicates the variation in the mass, luminosity, and surface temperature of the rejuvenated Sun under massloss Scenario 2. Since in this situation it is the ratio L/d2 that remains constant, Earth’s orbital radius d(AU) is also shown in the table. 158 Rejuvenating the Sun and Avoiding Other Global Catastrophes Table 5.3. Rejuvenated Sun evolution under mass-loss Scenario 1, where L(X) = constant. See Note 10 for an outline of how the Sun’s temperature evolution was determined. The mass-loss parameter N1 is described in Appendix A. X N1 M(X)/M T (K) 0.7 269 1.00 5780 0.6 286 0.82 5820 0.5 305 0.67 6020 0.4 326 0.53 6270 0.3 352 0.42 6580 0.2 381 0.32 7000 0.1 415 0.24 7680 0.01 453 0.18 9740 The evolutionary tracks corresponding to the two massloss scenarios are shown in Figure 5.2, along with a canonical solar evolution track. The rejuvenated Sun’s evolutionary path is initially upward, along the main-sequence in the second mass-loss scenario, where L/d2 is kept constant. In mass-loss Scenario 1, where the luminosity is kept constant, the evolution is simply towards higher surface temperatures with time. The outward expansion of Earth’s orbital radius in the second mass-loss scenario (last column of Table 5.4) results in its final location being some 3.4 AU from the Sun. As required by the conservation of angular momentum the orbits of the other planets and the main-belt asteroids will also increase by a factor of about 3.4 over their current values. Table 5.4. Rejuvenated Sun evolution under mass-loss Scenario 2, in which L/d2 = constant. See Note 10 for an overview of how the Sun’s temperature evolution was determined. The mass-loss parameter N2 is described in Appendix A. X N2 M(X)/M L(X)/L T (K) d(AU) 0.7 192 1.00 100 5780 1.0 0.6 204 0.87 13 5980 1.2 0.5 218 0.75 18 6200 1.3 0.4 234 0.64 25 6460 1.6 0.3 252 0.54 35 6800 1.9 0.2 273 0.44 44 7240 2.1 0.1 297 0.36 77 7970 2.8 0.01 324 0.30 114 10,060 3.4 Rejuvenating the Sun 159 Figure 5.2. The Hertzsprung-Russell diagram for the Sun’s evolutionary tracks according to various mixing and mass-loss scenarios. The dotted line corresponds to the Sun’s standard (canonical) evolutionary track, and the filled squares indicate points 2 and 3 from Table 3.2 (see also Figure 3.10). The filled circles on the main-sequence line indicate the luminosity and temperatures for a range of stellar masses. The large circle indicates the present main-sequence location of the Sun. The UV Problem Although it is the Sun’s total energy output per unit time (its luminosity) that is important with respect to Earth’s climate, the energy radiated by the Sun into space in a specific wavelength region is determined by the Sun’s temperature. Indeed, Wien’s law tells us that the wavelength ( max) at which a blackbody radiator emits its greatest energy flux varies with the temperature (T) in such a way that the product max T is a constant (= 2.8978 x 10−3). For the present Sun, with a temperature T= 5780 K, the greatest energy flux is at a wavelength of max ≈ 501 nm, which is why the Sun appears to be a yellowish-orange color to our eye. If a star, however, has a temperature twice that of the Sun, then max ≈ 250 nm, and this falls in the ultraviolet (UV) part of the electromagnetic spectrum. The increased temperature-related shift of max to the ultraviolet for a rejuvenated Sun is a potentially serious problem for future inhabitants of Earth. The UV segment of the electromagnetic spectrum falls in the wavelength region between 280 and 400 nm. As a result of atmospheric absorption it is mostly UV-A radiation, with 315 < 160 Rejuvenating the Sun and Avoiding Other Global Catastrophes (nm) < 400 that reaches Earth’s surface. It is UV-A radiation that enables the skin to generate vitamin D, but more harmfully it is also responsible for producing sunburns and cataracts. The lower flux of UV-B radiation, with 280 < (nm) < 315 that reaches Earth’s surface, however, is far more problematic since this radiation can cause direct damage to living cells, resulting in the development of cancerous growths. If the Sun’s temperature increases by a factor of two, as is indicated by the rejuvenation processes, its total energy output in the UV part of the electromagnetic spectrum will increase by a factor of about 40. Indeed, the UV-A flux will increase by a factor of 34, while that of UV-B increases by a factor of 66. For the rejuvenation process proceeding with L(X) = L (mass-loss Scenario 1), this is definitely a problem, since the increased UV flux will influence both Earth’s atmospheric structure and chemistry in addition to it being potentially deadly to all surface-dwelling life forms. Shielding Earth from the enhanced UV flux will be essential if the Sun is rejuvenated through mass-loss Scenario 1. In principle, however, the required shielding could be achieved by placing a large solar sail (see Figure 2.15) between Earth and the Sun. In the rejuvenation process in which L/d2 remains constant, Earth gradually moves outward in its orbit around the Sun. In this situation the flux of UV radiation reaching Earth is reduced by a factor proportional to d2, resulting in a much more modest increase in the received UV flux. This smaller UV flux increase tends to favor the solar rejuvenation process that keeps L/d2 = constant (mass-loss Scenario 2) over that which keeps L(X) = constant (massloss Scenario 1). The Extended Solar System Lifetime The solar rejuvenation process not only combats the natural aging of the Sun, quashing its bloated red-giant stage, it also extends the Sun’s lifetime. Eternal youth, however, cannot be gained and, eventually, the Sun will become a helium-rich white dwarf. How much additional lifetime the Solar System will gain through rejuvenating the Sun, however, will depend upon the exact amount Rejuvenating the Sun 161 of mass lost, the amount of chemical mixing, and the quantity of non-thermal pressure support that can be engineered. The main-sequence lifetime of a fully mixed star can be compared to that of a similar mass star evolving in the canonical fashion, and it transpires11 for Sun-like stars that in the zero massloss situation TH ∼ TMS, where TH is the hydrogen-burning lifetime of the fully mixed (homogeneous) star. Complete mixing alone, therefore, doesn’t greatly extend the lifetime of the Solar System. This result comes about because a fully mixed star evolves at a higher luminosity than a non-mixed star. In this manner, while the mixed star has access to more fuel in the form of hydrogen, it consumes it more rapidly. Indeed, this is the reason why mass-loss must accompany the additional mixing being invoked in the solar rejuvenation process. The mass-loss causes the star to evolve at a lower luminosity (recall Figure 5.1), and it is this effect – combined with the greater fuel ‘access’ – that results in the Sun achieving a greatly enhanced hydrogen-burning lifetime. If the mass-loss rate is assumed to be constant throughout the hydrogen-burning phase (as shown in Appendix A), then the ratio of the final mass to the initial mass of a fully mixed star is Mf/M0 = m(X = 0) = exp[ − N ( Q/c2 ) X0 ], where Q = 0.007 c2 is the energy liberated per kilogram of stellar material by the PP chain of fusion reactions. The effect of mass-loss and homogenization on the hydrogen-burning phase of a Sun-like star12 is shown in Table 5.5 Table 5.5. Hydrogen-burning lifetime improvement (TH/TMS) due to mass-loss and complete chemical mixing. The (assumed constant) mass-loss rate is parameterized according to N (first column), and the final to initial mass ratio is shown in the last column. N TH/TMS Mf/M0 25 1.1 0.88 50 1.2 0.78 100 1.4 0.61 150 1.6 0.48 200 2.0 0.38 250 2.6 0.29 300 3.5 0.23 400 7.5 0.14 450 12.2 0.11 162 Rejuvenating the Sun and Avoiding Other Global Catastrophes Table 5.5 indicates that for the typical mass-loss values required to rejuvenate the Sun, its main-sequence lifetime might reasonably be extended by a factor of between 4 and 6. The increase in the main-sequence lifetime for a homogenized Sun-like star undergoing mass-loss is, with reference to Table 5.1, about the same as introducing a 15 to 20 percent non-thermal pressure support within its interior. Mixing It Up Energy is transported by bulk convective motion in the outer third of the Sun’s radius, and in this region the material is well mixed and essentially homogeneous. The inner two-thirds of the Sun’s radius, however, is not compositionally mixed at the present time. The problem that the future star engineer must solve, therefore, is clear: How might the inner regions of the Sun be chemically mixed? Huber Reeves,13 in his book Atoms of Silence, captures the essence of the sought-after process by arguing that a pump of some type is required to cycle the hydrogen in the Sun’s envelope to the central regions, where it can be consumed through fusion reactions. Even more prosaically Richard Cathcart14 comments that the Sun must be stirred “much as one stirs a cup of coffee to mix the sugar and the liquid.” Although the various analogies and the result required can be stated in a clear fashion, the actual physics of the problem are profound. There is simply no easy way to mix the interior of the Sun. Hubert Reeves notes that additional mixing would take place within the Sun if a ‘hot spot’ could be created just above its fusion core. Indeed, such a process would work. However, the problem is, how might it be achieved? As shown in Chapter 3 the Sun’s fusion core extends over the inner third of its radius (a region actually containing ∼70 percent of the Sun’s total mass), so the asteroengineer has to find a means of transporting energy deep into the Sun’s interior. Directing asteroids or even large Kuiper Belt objects to impact upon the Sun will probably not produce the desired heating at the required depth. A numerical study carried out by David Andrews15 of Armagh Observatory, Northern Ireland, indicated that a 14-km diameter asteroid (with a mass ∼5 x 1015 kg) Rejuvenating the Sun 163 impacting the Sun’s surface with a velocity of 600 km/s might penetrate to a depth of 20,000-km before becoming thermalized with the stellar surroundings. This penetration depth, which is of order R/35, is 10 times smaller than required. A larger asteroid will penetrate somewhat further, but not by an appreciably larger amount. Ever resourceful and inventive, Reeves suggests that a highpowered laser might be used to generate a hot spot within the Sun, but again the problem is to get the energy to the right location deep within the interior. An alternative approach might be to develop methods to protect the intended impactor from being destroyed too soon. A combination of extremely high heat-resistant materials and strong enveloping magnetic fields might do the job, but the extent to which such futuristic technologies might be developed is hard to predict and currently unknown. Black Hole Mixing Black holes tend to be thought of as solely destructive entities and, indeed, they are at the heart of some of the most powerful and violent radiation-emitting sources that astronomers have ever detected. The million solar mass and larger black holes that power quasars and active galactic nuclei, however, are the leviathans of their race; the ones that the star engineer might be interested in are just a few hundredths of a millimeter in scale and weigh in at a relatively puny 1021 kg (about one one-hundredth the mass of our Moon). We know that millions of solar mass black holes do exist; it is also reasonably certain that stellar-mass black holes exist (such as in the X-ray binary system Cygnus X-1). But there is currently no clear observational data or consensus among astronomers concerning the existence of sub-stellar mass black holes. At this stage we shall assume that such black holes do exist, and that our distant descendants will work out ways of detecting, capturing, and transporting them through the solar neighborhood. The question now is what happens to a low-mass black hole when carefully placed at the surface of a star? Simply put, it will begin to oscillate backwards and forwards across the star’s interior. 164 Rejuvenating the Sun and Avoiding Other Global Catastrophes Displacement Speed Host star Time BH Figure 5.3. Displacement and velocity variation (scaled respectively to Rstar and Vmax) with time for a low-mass black hole placed at rest into the outer layer of a star. The solid line shows the variation of r/Rstar, while the dotted line shows the value of V(r)/Vmax. The period of oscillation is determined according to the distribution of mass within a star’s interior. It will move inward from its initial release point, gaining velocity as it moves toward the stellar center. It will pass through the center of the star (where it will attain its maximum velocity; see Figure 5.3), and thereafter it will begin to slow down as it journeys once again outward, just reaching the stellar surface at a location diametrically opposite to its release point. The whole oscillatory process will then repeat itself. Figure 5.3 illustrates the idea. The motion of a black hole placed at the surface of a star is described (at least to a first approximation) by the same equation that accounts for the small oscillations of a simple pendulum. The variation in the displacement and velocity of such a simple harmonic oscillator (sho) are described by the equations: r(t) = Rstar cos( t), and V(t) = Vmax sin( t), where t is the time since release, r(t) is the displacement from the center [with r(0) = Rstar, where Rstar is the radius of the star] , V(t) is the velocity [with V (0) = 0], Vmax is the maximum velocity, and is the angular frequency.16 Straightforward analysis reveals that the angular velocity is dependent upon the surface gravitational acceleration and the amplitude of motion. Accordingly, = (g/Rstar) 1/2, where g = G Mstar/(Rstar) 2 is the surface gravity of the host star (G is the universal gravitational constant). The period of oscillation Tsho is expressed according to this relationship: Tsho = 2 / . For the Sun, g = 273.76 m/s2 and Rstar = 6.96 x 108 m, which dictates that Tsho ≈ 104 seconds (corresponding to about two and three-quarter hours). Now, the result just presented is derived from the assumption that the Sun has a uniform and constant density throughout Rejuvenating the Sun 165 its interior; a more detailed calculation16 allowing for the Sun’s central mass concentration, however, reveals that the black hole oscillation time is, in fact, about 70 percent of Tsho, indicating an oscillation period of just under two hours duration. A low-mass black hole moving through a star just once won’t produce much of a mixing effect. In this engineering project, however, time is not considered to be of the essence, and the black hole will continue to oscillate back and forth across the star hundreds of millions of times, with each oscillation stirring the interior just a little bit more. Mixing will be initiated in the wake of the black hole once its speed of motion exceeds that of the local gas sound speed, since under such conditions a shockwave will be produced, and it is this effect that will produce the required interior mixing. An everyday analogy of the effect being envisioned (without the shockwave) is that of placing a small ball-bearing inside of a spray can of paint. By shaking the can the ball-bearing ‘stirs’ the paint and keeps it well mixed and fluid. A detailed set of calculations16 indicates that a black hole placed within the Sun will be traveling at a maximum speed of something like 1,600 km/s when it reaches the Sun’s center—a speed that is three times greater than the local sound speed. Indeed, the speed of a black hole oscillating within the Sun exceeds that of the local sound speed over 95 percent of its interior, only traveling at subsonic speeds in the outer 5 percent by radius. A Steady Stellar Diet With respect to reducing the mass of a star there are essentially two scenarios that might be applied. The mass can be either extracted from the surface, or it can be extracted from the deep interior. The first method essentially entails the inducement of an enhanced stellar wind, while the latter invokes accretion onto a central black hole. A number of solar-mining and star-lifting scenarios have been described over the years, but here we will review just two of them. The first method has been described by Paul Birch17 and involves a fleet of ramscoops. The second method was developed by David 166 Rejuvenating the Sun and Avoiding Other Global Catastrophes Criswell18 and involves the placement of multiple particle accelerators in orbit around the Sun. The ramscoop method of solar-mining is illustrated in Figure 5.4. It was originally developed by Birch to extract just the heavy elements – or metals – from the Sun and, accordingly, the hydrogen and helium is not specifically collected. The ramscoop method could be extended, however, along the lines of the Bussard Ramscoop Hyperbolic trajectory Sun Passage through Sun’s outer atmosphere Direction of motion Swept up atmospheric gas Storage area for extracted ‘metals’ H, He & ‘metals’ H and He ‘exhaust’ RL Radius of gyration Magnetic field line Ramscoop Figure 5.4. Solar-mining by the ramscoop method. The ramscoop is accelerated to skim through the Sun’s outer atmosphere (upper figure). Ionized material entering the ramscoop’s mouth will interact with a strong internal magnetic field and begin to move along spiral trajectories (center figure). The radius of the spiral trajectory is governed by the mass of the ion, and the more massive particles will have larger Lamor radii (RL) (lower figure). Rejuvenating the Sun 167 ramjet19 to sweep up the hydrogen and helium in the Sun’s outer atmosphere. The mining scenario begins with the ramscoop being accelerated to a fast (hyperbolic) orbit that allows it to skim through the Sun’s outer atmosphere, thereby gathering material into its ‘mouth.’ The ionized material brought into the ramscoop’s interior will interact with a strong, internally generated magnetic field, causing it to move along spiral trajectories. The radius of the spiral path is described by the Lamor or gyration radius, which increases in proportion to the mass of the trapped ion. In this manner the various ions can be sorted, one mass from another. By exploiting the ion-mass segregation process the various solar ‘metals’ can be siphoned off as they spiral along the magnetic field lines on their way to the ramscoop center. Having passed through the Sun’s outer atmosphere, the ramscoop returns to orbit with a diminished speed and an increased mass. Following a docking maneuver with an orbital storage ship, the ramscoop is unloaded and once again accelerated towards the Sun for another mass-extraction dive. The mass-loss procedure described by David Criswell18 is illustrated in Figure 5.5. In this mass-loss scenario a constellation of satellite accelerators are placed in a polar orbit around the Sun. Counter-directed and oppositely charged ion beams are then sent between each of the accelerators. In this fashion, the beams produce a loop of current around the Sun. By causing the plane of the accelerators (and the current loop) to rotate around the Sun’s spin axis, two channels will be formed through which the Sun’s outer atmospheric plasma will be continuously ejected. In this scenario the material will be predominantly swept outward in the plane of the Sun’s equator. An alternative scenario to that illustrated in Figure 5.5 has the accelerator satellites orbiting the Sun’s equator, and in this configuration the current loop will cause material to be ejected from the Sun’s poles. To induce mass-loss under this scenario, however, the polar regions need to be heated so the atmospheric atoms will have enough energy to escape the Sun’s gravitational potential. Additionally, the orbital radius of the accelerator satellites could be made to systematically contract and expand, heating the Sun’s atmosphere through the generation of sound waves. Criswell calls this mass-loss mechanism the “huff-n-puff” method. 168 Rejuvenating the Sun and Avoiding Other Global Catastrophes Sun’s spin axis Accelerator satellites in a common polar orbit Plasma flow B Counter-directed and Thrust ⇑ oppositely charged ion beams exchanged between accelerator satellites Sun’s spin axis Sun’s equator ⇑ Thrust B Plasma flow Figure 5.5. Equatorial mass ejection from the Sun. A constellation of accelerator satellites are placed into a common polar orbit around the Sun (upper left). By exchanging ion beams between each satellite a current loop BB is established (lower right). By rotating the plane of the accelerator orbit, surface solar material is driven outward along two ‘exit’ channels. (Diagram adapted from Criswell.)18 Martyn Fogg20 has noted that, in principle, by varying the relative polar mass-loss rates a star might literally be lifted and propelled by a polar mass-ejection mechanism (Figure 5.6). Clearly the star-lifting scenario will have an important role to play in the re-direction of rogue stars (such as Gliese 710) away from a direct intercept path with the inner Solar System or our Solar System’s Oort Cloud (see Figure 2.7). Rejuvenating the Sun 169 Thrust Magnetic field Current loop Mass outflow Energy beam Figure 5.6. Lifting a star by the generation of an equatorial current loop. The star receives a thrust in the direction opposite to that of the material being ejected from the actively heated pole. (Diagram adapted from Fogg20 and Criswell.)18 Brave New Worlds To simply let the material extracted from the Sun dissipate into space would be a great waste of a useful resource. Although it is difficult to envision how the process might work, Criswell18 has suggested that our remote ancestors might try to take the Sun apart layer by layer. If this approach can be made to work, lowmass dwarf stars could be constructed. As described in Chapter 3, the minimum amount of hydrogen that is required to produce a bona fide star is about 0.1 M Following a solar rejuvenation scheme similar to the one outlined above, the future star engineer might, therefore, attempt to produce five to six dwarf stars as the Sun is modified. Such solar sibling stars would provide an immense wealth of additional energy for our descendants to utilize. By constructing Dyson sphere complexes around such sibling stars they could be turned into vast manufacturing centers or the parent stars to O’Neill-style space colonies. 170 Rejuvenating the Sun and Avoiding Other Global Catastrophes The most likely region to assemble or eventually place (via star-lifting) solar dwarf star siblings will be in the inner Oort Cloud region. If a dwarf star of mass 0.1 M is going to have a small to negligible effect on the orbits of objects out to, say, 1,000 AU from the Sun, then it must be placed in a solar orbit of radius ∼1,500 AU. Such an orbit will clearly have a dramatic effect on the cometary nuclei in the inner Oort Cloud, but it is presumably safe to assume that by this advanced stage of engineering development, cometary impact avoidance will be a problem that has long been solved. A possible configuration of an orbit21 for a full set of six solar sibling dwarf stars is shown in Figure 5.7 In the configuration shown each dwarf star leads and follows its nearest companions by 60 degrees. The separation between each dwarf star and its nearest neighbor is √3 R, where R is the orbital radius around the Sun, and the maximum separation between any two dwarf star complexes is 2R. Such a configuration minimizes the travel and communication times between the habitats in orbit about each of the dwarf stars, as well as with the planets in orbit around the Sun. The main-sequence lifetime of a 0.1 M star is measured in trillions of years, and (as illustrated in Figure 3.6) the fully mixed interiors of such stars naturally ensure that they won’t undergo a red-giant phase (see Figure 3.11). Although we do not know how it might be achieved in practice, it seems that dwarf star production D5 D6 D4 D1 Sun R D3 D2 Figure 5.7. A possible arrangement of solar sibling dwarf stars. In this configuration the rejuvenated Sun has a mass of 0.4 M, and each dwarf star (D1 to D6) has a mass of 0.1 M (Note, M means the present mass of the Sun). Each dwarf orbits the Sun at a distance R and is separated from its nearest neighbors by a distance of (√ 3 R). Rejuvenating the Sun 171 and orbital husbandry should be a highly sought after end product of the solar rejuvenation process. An Alien Beast Within To some extent the notion of placing a black hole at the center of the Sun might sound like the scenario for the film Alien (20th Century Fox, 1979): “Lurking deep within the host star the black hole feeds from the inside outward, consuming the star inner layer by inner layer, ever hungry, until all has passed into its bloated maw.” Well, this all sounds rather dramatic, but there are in fact good reasons to consider placing a black hole at the center of the Sun in order to rejuvenate it. The possibility that the Sun might contain a low-mass central black hole was considered in some detail during the 1970s as a possible solution to the so-called solar neutrino problem.22 The basic principle at work in this scenario is that the Sun can tap the energy liberated by material as it falls into the black hole. The more energy supplied by the accreting black hole, the less energy the Sun has to generate through fusion reactions and the lower its central temperature will be. The mathematical description of an accreting black hole placed within a star is given in Appendix B, and we refer the reader there for details. The main point for discussion at this stage relates to Equation (B.6), which describes star consumption time (literally, the time for the black hole to consume the star) Tconsume to the canonical main-sequence lifetime. If we once again assume that sub-stellar mass black holes with Mbh ∼1021 kg exist, and that such objects can be recruited in the Sun rejuvenation process, then Tconsume/TMS ∼0.1 when the energy conversion efficiency (see Equation B.1) is 10 percent. This result tells us that in the case of the Sun, Tconsume is of order one billion years. Even if the black hole is inserted into the Sun near to the end of its canonical main-sequence lifetime it will not buy our distant descendants much additional time. If a highly optimistic energy conversion efficiency of, say, 25 percent is assumed, then the consumption time for the Sun is increased to about 2.5 billion years. 172 Rejuvenating the Sun and Avoiding Other Global Catastrophes To increase the lifetime of the Sun well beyond its canonical main-sequence lifetime, it would seem that the only option for the star engineer is to develop techniques for controlling the accretion rate onto the black hole. The luminosity of a black hole radiating at its maximum rate is set by the Eddington limit,23 which is related to the mass of the black hole: (LEdd/L = 4.3 x 104 (Mbh/M From this relationship we see that once the black hole mass exceeds 2 x 10−5 M it will already supply enough energy to power the Sun at its current output level. A 1021 kg black hole will achieve a mass of 10−5 M some 400 million years after its insertion into the Sun (assuming 10 percent energy conversion efficiency). After this time the accretion rate would have to be carefully controlled and reduced so that Lacc = L rather than Lacc = LEdd. The Sun’s central region is certainly an extreme environment from which to expect any Earth-produced machinery to work, but we are in the luxurious position of saying that this is a problem for our distant descendants to solve. Although the controlled accretion onto a black hole at the center of the Sun can, in principle, quash the requirement for nuclear energy generation, the first problem to be solved is the placement of the black hole at the Sun’s center. To a good initial approximation, however, the black hole placement can be modeled as a harmonic oscillator with an additional damping (or resistive) term. Under this approximation it can be shown16 that the equation of motion has a straightforward solution with displacement r(t) being described by the equation r(t) = Rstar e−Dt cos( t), where D is a constant and is related via various constants to the angular frequency defined earlier for the simple harmonic oscillator. The constant D in the exponential term is a damping parameter that reduces the amplitude of oscillation. The larger the value of D the more rapidly the black hole will settle toward the center of the Sun. The timescale upon which the amplitude of motion decreases is given by the e-folding timescale defined as Te−fold = 1/D. For each time interval corresponding to Te−fold the amplitude variation decreases by a factor of e ≈ 2.718. If the damping of the black hole’s motion is solely due to accretion, then it turns out that the e-folding time is of order many tens of millions of years. Such a settling timescale may be thought far too slow24 by future star engineers and, consequently, steps Rejuvenating the Sun 173 Trelease Amplitude reduced by a factor of e Te-fold Capsule displacement Host star Capsule BH Time Figure 5.8. Delivering a low-mass black hole to the center of a star. This schematic diagram shows the black hole initially contained within a capsule designed to produce a large damping effect. To the right the displacement of the capsule with time is indicated. The amplitude of oscillation decreases by a factor of e = 2.718… on intervals corresponding to Te-fold. The black hole is released at time Trelease after which it begins to accrete material from the host star. could be taken to make the damping term D as large as possible, thus reducing the e-folding time. One way in which this could be achieved is to initially constrain the black hole to reside within a delivery capsule that has been specifically designed to produce a significant viscous damping effect (Figure 5.8). High damping might be produced, for example, by the delivery capsule having a large surface area. An everyday analog to this scenario is that of a time-release medication capsule. In this fashion the capsule rapidly settles to the center of the Sun on a timescale of perhaps hours to days.16 Once it has settled to the center the capsule releases the black hole in place, and the accretion of stellar material can begin. Solar Wrap Control A solar wrap might be thought of as a shrunken Dyson sphere. Rather than having a radius of order 1AU, a solar wrap, as the name suggests, would envelop the Sun at a distance of about 10 solar radii. In addition, and in contrast to a Dyson sphere, a solar wrap is designed to reflect most of the energy it receives from the Sun back toward its photosphere. The essential idea of the wrap is to cause the Sun to heat its own outer layers. This is a recipe in which the Sun essentially cooks itself. By heating its outer layers, the Sun will ultimately be forced to reduce the amount of energy it generates within its interior. Indeed, if none of the Sun’s energy 174 Rejuvenating the Sun and Avoiding Other Global Catastrophes is allowed to escape into space (an extreme example that isn’t very practical, perhaps), it will eventually evolve an isothermal interior with no energy generation taking place within its core. This perhaps seems an odd state of affairs, but the argument builds upon the results discussed in Chapter 3 and the summary shown in Figure 3.12. The point, of course, is that if the Sun isn’t generating energy within its core, then it isn’t using up its hydrogen fuel supply, and as long as the wrap stops energy leaking out into space it will have an extended, potentially indefinite, lifetime. The key concept here is that a star only generates energy within its interior because it loses energy into space at its surface. Plug the surface ‘leak’ and the requirement to be hot enough at the center to fuse hydrogen into helium goes away. The solar wrap scenario works by exploiting the negative feedback mechanisms that operate within a star. Recall from Chapter 3 that the stability of a star is maintained by the existence of a temperature gradient; the temperature at each layer inside a star is just right to provide the correct gas pressure to support the weight of overlying layers. Now, if the energy radiated at the surface of a star is stopped from escaping into space by a totally reflecting cover (i.e., a solar wrap), a number of effects will come into play. First, the outer layers themselves will be heated and will consequently expand somewhat. Second, the heated region will gradually advance inward, forming a constant temperature (isothermal) zone. An outer isothermal zone comes about because there is no requirement to transport energy outward. When a region is isothermal there is no temperature gradient (it is the same temperature throughout), so the star will have to reorganize its internal structure so that the density gradient can maintain the stability condition. Recall that the ideal gas equation indicates that the pressure is determined by both the temperature and the density. Eventually the isothermal region will extend all the way to the center, and the star will adjust its internal density at each interior layer so that it remains stable. The temperature of the isothermal star will be adjusted downward until it is just below the threshold for nuclear fusion reactions to occur. It is in this manner that the effect of a solar wrap becomes clear. By being surrounded by a fully reflective cover, the Sun will eventually evolve into a stable, Rejuvenating the Sun 175 isothermal object requiring no internal energy generation, this later condition coming about because there is no energy lost into space. The negative feedback mechanisms described in Chapter 3 will still apply to the isothermal Sun and, in principle, it will remain in a dormant state until some energy is allowed to escape through the wrap. At this point the Sun will undergo an internal readjustment so that the central temperature is raised by just the right amount to enable nuclear fusion reactions to replenish the energy that has been lost into space. A series of detailed calculations25 indicate that if, for example, the Sun can be transformed into an isothermal state, with a constant internal temperature of 106 K (i.e., below the temperature for efficient fusion reactions) and a central density somewhere within the range 100 to 1,000 kg/m3, then it will expand by a factor of about 5 ½ times its present radius. In relation to a Dyson sphere, therefore, a solar wrap is not especially large, being in comparison some 40 times smaller in radius. A fully reflective solar wrap is clearly not much use, of course; the Sun still has to provide energy to heat the planets within the Solar System. This suggests that an actively maintained (or leaky) wrap will have to be developed such that controlled amounts of energy are reflected back into the Sun’s outer layers as needed. If such a Sun-altering scenario can be made to work, and if it is to extend the Sun’s hydrogen-burning lifetime, then it has the added bonus that the Sun’s mass is not changed and, consequently, there is no migration of planetary orbits. The dynamic solar wrap scenario won’t necessarily stop the red-giant stage from occurring. But slowing down the rate at which the Sun converts hydrogen into helium will extend the time interval over which life can survive in the inner Solar System. What the Future Holds If humanity is to have a long-term future, then it must tame the Sun. There has been a long tradition of simply assuming that the aging of the Sun will destroy Earth and drive humanity (or – as is more likely the case – a very, very small and elite subset of humanity) into interstellar space to drift like a race of 176 Rejuvenating the Sun and Avoiding Other Global Catastrophes landless gypsies hoping to find a new planet to colonize. This may, of course, come to pass, but it seems that this would be a failed future. That is, it fails humanity collectively and deprives countless billions of people a potentially rich and prosperous life within a Solar System full of resources. In this chapter we have attempted to outline a number of possible options for rejuvenating the Sun. Some of the scenarios are more fanciful than others, but the key point here is that there are conceivable options and possibilities. The Sun can be tamed, and it is not inevitable that it will become a red-giant. Clearly, we do not know what will come to pass in the future, and there may be many other possible means of rejuvenating the Sun; who knows what our descendants 1 million, 10 million, and 100 million years from now will be able to do. The future holds great promise for humanity—provided that humanity is prepared to realize it. Notes and References 1. James Lovelock, The Revenge of Gaia, Penguin Books, London (2006). pp. 44–45. Lovelock further argues that in about 1 billion years’ time the Sun’s luminosity will have reached a level 9 percent greater than it is now, and that Gaia, the self-regulating ‘living’ world-system, will die. Indeed, Lovelock argues that in as little as100 million years from now the Sun’s energy output will have risen to a level that will force the biosphere into a new ‘hot-state.’ 2. From Equation (3.12) it can be seen that the luminosity at the beginning and the end of the main-sequence requires that the mass be reduced by a factor [( begin/ final) 75/(1 + X0)]1/5 = 0.286, given that begin = 0.613 and final = 1.613. 3. See Beech, M., Sensitivity in the models of massive stars. Astrophysics and Space Science, 161, 133–143 (1989); Beech, M., and Mitalas, R., The homogeneous evolution of massive stars. Astronomy and Astrophysics, 213, 127–132 (1989). 4. In this derivation we have made use of the binomial expansion, which for small values of gives (1 + ) −1 ≈ (1 – ). 5. George Greenstein. Order of magnitude ‘theory’ of stellar structure, American Journal of Physics, 55 (9), 804–810 (1987). 6. Martin Beech. Aspects of an asteroengineering option. Journal of the British Interplanetary Society, 46, 317–322 (1993). Rejuvenating the Sun 177 7. For further details see M. Beech, A novel stellar model: ‘a sacrifice before the lesser shrine of plausibility’. Astrophysics and Space Science, 168, 253–261 (1990). 8. The helium main-sequence is nicely described in the book by R. Kippenhahn and A. Weigert, Stellar Structure and Evolution, Springer-Verlag, Berlin (1990), pp. 216–218. Detailed calculations reveal that a 1M chemically homogeneous helium-burning star is about 300 times more luminous than a 1M chemically homogeneous hydrogen-burning star. Helium-burning stars have smaller radii and higher central and surface temperatures than hydrogenburning stars of the same mass. The very high central temperature appropriate to helium-burning via the triple- reaction (see Note 15, Chapter 3) dictates that the energy is carried outward from the center by convection rather than radiation in homogeneous, solarmass helium-burning stars. 9. This point is further discussed in the research paper by M. Beech and R. Mitalas, Effect of mass-loss and overshooting on the width of the main-sequence of massive stars. Astrophysical Journal, 352, 291–299 (1990). 10. The Stefan-Boltzmann law is used to determine the surface temperature of the Sun. Accordingly, Te4 ∼L/R2, where L is the luminosity and R is the radius. Now, Equation (3.5) indicates that R ∼M/Tc, where M is the mass of the star and Tc is the central temperature. To evaluate the central temperature we have solved the integral: L = dm, where is the energy generation rate per unit mass of stellar material, as given by Equation (3.11). Accordingly, L ∼ X2 T c 6+ / 2 Typically, for solar-mass stars, ≈ 4. Working through the substitutions, the surface temperature of the star can be shown to vary as Te X /Te X0 = l 1+2 /4 m−1/2 1+x −2 X− Where = 1/ +6 . 11. Following Reference 7, TH (x) ∼TMS [1 + x17] (1 + X0)/(17 q), where q ∼0.1 is the mass fraction of hydrogen available to a star evolving canonically. At hydrogen exhaustion, x = 1/(1 + X0), and [1 + x17] ∼1, leaving TH ≈ TMS (1 + X0)/1.7, and since X0 ≈ 0.7 so, TH ≈ TMS. 12. Building upon Reference 7, the main-sequence lifetime of a homogeneous Sun-like star with mass-loss is THML = TMS [(1 + X0)/q] exp(4 k) x17exp(− 4 k x) dx, where the integral is from x to 1, and k is given in Equation (A.7). 13. H. Reeves, Atoms of Silence: An Exploration of Cosmic Evolution. MIT Press, Cambridge, Massachusetts (1985). 14. From the essay by R. B. Cathcart, True asteroengineering: intrusive sun-stoking rejuvenation macroprojects. See http://www. daviddarling.info/Cathcart.html. 178 Rejuvenating the Sun and Avoiding Other Global Catastrophes 15. A. D. Andrews, Investigations of micro-flaring and secular and quasiperiodic variations in dMe flare stars. Astronomy and Astrophysics, 245, 219–231 (1991). 16. In the detail calculation we have assumed that the Sun has an n = 3 polytropic structure [see Kippenhahn and Weigert’s book, Reference 8], and the equation of motion for the black hole is integrated numerically. The polytropic approximation allows for the mass distribution to be centrally condensed with, in the n = 3 case, the central density being some 54 times greater than that of the constant density model of the same mass. 17. Paul Birch, Supramundane planets. Journal of the British Interplanetary Society, 44, 169–182 (1991). 18. David Criswell, Solar system industrialization: implications for interstellar migration. In Interstellar Migration and the Human Experience. R. Finney and E. Jones (eds.), University of California Press, Berkeley (1985). pp 50–87. 19. An interstellar ramjet (or ramscoop) was described by physicist, Robert Bussard in his paper, Galactic matter and interstellar flight, Astronautica Acta, 6, 179–194 (1960). The spacecraft described by Bussard uses a large scoop for channeling interstellar material into a central ‘fusion’ chamber, where energy is generated to propel the ship. The faster the spaceship goes, the more interstellar material it can sweep up and, consequently, the spacecraft could, in principle, reach relativistic speeds. 20. Martyn Fogg, Solar exchange as a means of ensuring the long-term habitability of Earth. Speculations in Science and Technology, 12 (2), 153–157 (1988). Viorel Badescu and Richard Cathcart, Stellar engines for Kardashev’s Type II civilizations. Journal of the British Interplanetary Society, 53, 297–306 (2000) have introduced the term Class A engines for machines (processes) that ultimately produce a thrust force from a star. Class B engines, on the other hand, use the radiation emitted by a star to produce mechanical power. A Class C engine is a combination of both A and B, and Badescu and Cathcart argue that such machines will provide a Kardashev Type II civilization with both power and interstellar transport (such as in the gargantuan, Solar System-shifting sail processes envisioned by Leonid Shkadov. 21. Aesthetics more than anything else underlies the configuration shown in Figure 5.8. The system, however, is stable provided that the mass of each sibling star is the same. Indeed, each sibling will move around the circular orbit (circumscribed about the hexagon) in pace and equally spaced from its neighbors. In essence, the ‘hexagon’ (with a sibling star at each node) rotates about the center (the Sun) as Rejuvenating the Sun 179 if it were a ridged structure. It can be shown, in fact, that any regular polygonal arrangement of equal mass dwarfs will follow a stable circular orbit about the central Sun. Three sibling dwarfs of mass 0.2 M situated at the ‘corners’ of an equilateral triangle centered on the Sun would also be a possible stable configuration. For a good review of such orbits, see Eugen Butikov, Regular Keplerian motions in classical many-body systems. European Journal of Physics, 21, 1–18 (2000). 22. John Bachall [Neutrino Astrophysics, Cambridge University Press, Cambridge (1989)] nicely summarizes the ‘non-standard’ stellar evolution solutions offered to solve the solar neutrino problem. The solar neutrino problem has now been resolved in terms of ‘new physics,’ relating to the flavor-changing behavior of neutrinos. 23. The Eddington luminosity is derived on the basis that the pressure support for a star is that provided by radiation alone: P = Prad = (1/3) aT4, where a is the radiation constant and T is the temperature. Accordingly, LEdd = 4 GcM/ , where is the opacity of stellar material. At the high temperatures required for radiation pressure to provide the support for a star, the electron scattering opacity dominates, and the Eddington luminosity becomes LEdd/ L = 4.3x104 (M/M. 24. In this scenario the black hole is not required to mix the star and, accordingly, the aim is to place it at the Sun’s center as quickly as possible. 25. In these calculations we have numerically integrated the differential equations describing the pressure and mass variation within a star as a function of radius. [See, i.e., Kippenhahn and Weigert’s book (Reference 8) for these equations.] Setting the temperature and central density as parameters to be chosen, the equations are integrated outward from the center to a radius R* at which distance a total mass of M* = M is enclosed. This procedure determines the size of the isothermal Sun for the chosen temperature and central density. I have assumed in the calculations that the equation of state is that of a perfect gas. The stability of the isothermal Sun models has been tested against the gravo-thermal catastrophe condition described by Donald Lynden-Bell and Roger Wood [The gravo-thermal catastrophe in isothermal spheres and the onset of red-giant structure for stellar systems. Monthly Notices of the Royal Astronomical Society, 138, 495–525 (1968)]. For the perfect solar wrap considered in these calculations, stability is assured provided the central density is less than ∼103 kg/m3 when T = 106 K. If a higher temperature of say 5 x 106 K can be engineered, then central densities between 104 to 105 kg/m3 are allowed, and the equilibrium radius is reduced to of order 2 R.