Resolution of the Faint Young Sun Paradox via the Expanding Earth and Radiation Balance Equilibrium Hypothesis

We present a plausible solution to the now forty seven year old paleoclimatology riddle 1 of the so-called Faint Young Sun Paradox via the combined hypothesis of the conservation of the 2 state of radiation balance between the Earth and Sun and that of an expanding Earth, where, in the 3 face of a changing (increasing) Solar luminosity, the Earth would maintain steady temperatures 4 by re-adjusting the height of its atmosphere. That is to say, depending on whether or not the 5 radius of the solid Earth is changing, this re-adjustment of the height of the Earth’s atmosphere 6 would mean two things — i.e.: (1) either the height increases — in which event the Earth accretes 7 matter from its immediate surroundings (i.e., the obvious pool formed by the Solar wind) thereby 8 increasing the mass of the Earth’s atmosphere, or: (2) the height decreases — in which event 9 the Earth naturally expels matter from its atmosphere, thereby decreasing the effective mass of 10 the Earth. We demonstrate that if — as the current state of the art ITRF observations seem to 11 indicate, namely that — the Earth’s landmass is steadily expanding globally at a paltry rate of 12 ∼ +0.45± 0.05 mm · yr−1, and, that the Earth’s atmosphere is to have a present radial vertical 13 height of about one third of the Earth’s radius (∼ 2860 km) from the Earth’s surface, then, one 14 can (might) with relative ease, explain not only the presence of liquid water on the Earth’s surface 15 some ∼ 3.20± 0.70 Gyr ago during the Archaean eon when the Sun was about 75% of its current 16 luminosity, but also the present radial expansion rate of the Earth. When all is said and done, 17 the Earth system is herein cast as an auto-self-regulating incubator where the auto-self-regulating 18 mechanism is as a result of the Earth’s atmosphere responding by automatically re-adjusting its 19


Introduction
Writing in 1972 in the journal Science, renowned American astronomer, cosmologist, 23 astrophysicist, astrobiologist, author, and one of the greatest inspirational science com- 24 municators in the history of humankind -Carl Edward Sagan (1934Sagan ( -1996, and his 25 fellow astronomer -George Mullen; brought (for the first time) to the international lime-26 light 1 , an apparent paradox [3] concerning the evolution of the Sun and the supposed 27 presence of liquid water on the Earth's surface [see e.g. Refs., [3][4][5][6][7][8]. As initially pointed 28 out by Donn et al. [2] and Sagan & Mullen [3] noted that according to the then just 29 established evolutionary stellar models that describe stars like our own Sun -models 30 that still hold to this day [see e.g. Refs., 9,10]; the Sun's energy output (which should 31 1 According to Feulner [1], it was Donn et al. [2] who was were the first -in recorded literature -to point out the apparent discrepancy between the low solar luminosity predicted for the young Sun and the evidence for liquid water on early Earth. Sagan & Mullen [3] where the first to bring this problem to the popular attention of the wider scientific community.

1.
Greenhouse Effects [see e.g., 20,21,23,30,31]. In this scenario, we have enhanced 70 greenhouse effect by carbon dioxide or methane, geothermal heath from an ini- 71 tially much warmer terrestrial core, a much smaller Earth albedo, life developing 72 in a cold environment under a 200 m thick ice sheet, a secular variation in the 73 gravitational constant, etc. According to Kasting [22], most of these greenhouse 74 effect models have serious shortcomings: for example, the greenhouse effect from 75 methane appears to be self-limiting, and not enough carbon dioxide is indicated 76 by the geological record to justify a greatly enhanced greenhouse effect in the past. 77 Further, according to Rossing et al. [5], examination of Archaean sediments appears 78 inconsistent with the hypothesis of high greenhouse concentrations. Rossing et al.
79 [5] argues that instead, the obtaining moderate temperature range of the Earth's 80 2 Archaean eon, also spelled Archaean eon, is a period in the Earth's history which began about 4.00 billion years ago with the formation of Earth's crust and extended to the start of the Proterozoic eon 3 2.50 billion years ago. During this time, unicellular organisms are the earliest forms of life that emerged.
system through the eons may be explained by a lower surface albedo brought about 81 by less continental area and the 'lack of biologically induced cloud condensation nuclei'. 82 This, Rossing et al. [5] say, would have led to increased absorption of solar energy, 83 thereby compensating for the lower solar output. 84 85 2. Astrophysical Influences [e.g., 5,18,32,33]. For example, Rossing et al. [5,18] hypoth-86 esizes a lower Earth albedo and this owing to considerably less continental area 87 (this may include an Earth with a smaller radius) and to the lack of biologically 88 induced cloud condensation nuclei. This would make an important contribution 89 to moderating surface temperature in the Archaean eon. Further, Rossing et al. [18] 90 suggests that the lower albedo of the early Earth provided environmental con-

3.
Active Young Sun Hypothesis [34]. Using κ 1 -Ceti as a comparison for the young 98 Sun, Karoff [34] have argued that not only was the young Sun much more effective 99 in protecting the Earth environment from galactic cosmic rays than the present day 100 Sun; it also had flare and corona mass ejection rates up to three orders of magnitude 101 larger than the present day Sun. By means of the Forbush Effect 4 , Karoff [34]  temperatures. For such a scenario, Iorio [14] calculates that, the change in the mean 125 Earth-Moon distance δr ⊕ (t)/r ⊕ (t) will have to be related to the change in the Solar 126 luminosity δL (t)/L (t), as follows: Compared to δL (t)/L (t), the term δr ⊕ (t)/r ⊕ (t) is four orders of magnitude 135 too small, the meaning of which is that the secular Earth-Moon system drift of 136 (7.00 to 15.00) cm · yr −1 , this can not be responsible for the sustenance of the liquid 137 water during the Archaean eon to the present day. If indeed a closer Earth is the 138 solution to the FYS-Paradox, then, an extra mechanism presently pushing the Earth-

139
Moon system away from the Sun is needed [14,15].

140
In-conclusion 141 Extensive reviews on this subject have been carried out with the most recent being 142 those by Iorio [14,15] and Feulner [1]. The FYS-Paradox not only remains an Open

144
In the present endeavour, we shall add a completely new solution to this long-standing where the Earth-system is cast as an auto-self-regulating incubator which auto-adjusts 147 its albedo as the Solar luminosity changes.

148
As said, the Archaean eon occurred during the period: t A ∼ (3.80 to 2.50) Gyr ago.

149
In-order for us to have convenient calculations, we need a single value of t A rather than 150 a lower and upper limit. To that end, we shall take the Archaean eon period: gives a change in the Earth radius of:

274
As is well known and as has already been stated -the mean global temperature of the To maintain this long-term temperature stability, the Earth must radiate into space a flux 282 of energy sufficient to just balance the input from the Sun -the meaning of which is 283 that, to a good degree of approximation, the Earth has been, and most probably is, in 284 radiative equilibrium with the Sun's radiation that it freely receives. This situation is 285 quite easy to fathom.

286
By absorbing the incoming Solar radiation, the Earth will warm up and as a con-287 sequence thereof, its temperature will rise accordingly. particularly so if they are surrounded by empty space. This radiation is referred to as the 297 outgoing radiation. As long as the incoming radiative flux is larger than the outgoing, 298 the radiation receiving object will continue to warm, and its temperature will continue 299 to increase according. This in turn will result in an increase in the outgoing radiation Now, in-order to compute this state of radiation balance between the radiant Sun 307 and the Earth system, let: L (t) = 4πσ 0 R 2 (t)T 4 (t), be the Solar luminosity at time t, 308 with σ 0 being the usual Stephan-Boltzmann constant, R (t) the Solar radius at time t, 309 T (t) the Solar temperature at time t; and let r ⊕ (t) be the mean distance of planet Earth 310 (or any given planet) from the Sun at time t. The total Solar flux: F (r, t), arriving at the 311 spherical shell of radius r centred about the Solar center is such that: Not all Solar radiation intercepted by the Earth is absorbed by the Earth system - Alkenones are long-chain unsaturated methyl and ethyl n-ketones produced by a few phytoplankton species of the class Prymnesiophyceae [e.g., 70] 7 The Ice Age is believed to be a period of long-term reduction in the temperature of Earth's climate, resulting in an expansion of the continental ice sheets, polar ice sheets and mountain glaciers. There are three main types of evidence for ice ages: geological, chemical, and paleontological.
by the Earth at time t is therefore [1 − A ⊕ (t)]. The effective power P abs ⊕ (t) absorbed by 320 the Earth system is therefore given by: According to the Stefan-Boltzmann law, the total energy emitted by the solid Earth per 322 unit area is given by ⊕ σ 0 T 4 ⊕ (t), where, ⊕ is the emissivity 8 of the solid Earth. The 323 emitting total area is the surface area of the solid Earth, 4πR 2 ⊕ (t), therefore, the total 324 energy emitted by the solid Earth per second is: This energy balance requires that (Incoming Radiation = Outgoing Radiation) so that 326 when averaged over eons, we will have: thus: This deceptively simple looking Eq. (9) This Eq. (10) can further be re-written so that it reads: where: S (t) = L (t)/4πr 2 ⊕ (t), is the Solar constant or Solar irradiance at time t, and this 334 important quantity is measured by satellites orbiting above the Earth's atmosphere at 335 1.00 AU, and its current accepted value is: 1360.80 ± 0.50 W · m −2 [76].

336
As can be read off from column (3) of for the other planets -Mercury, Venus and Mars, there exists a discrepancy between 343 theory and observations. As is the case with the Earth system, the reason given for this 344 discrepancy is that the derivation of Eq. (11) does not include all the processes at play.

345
All in an effort to improve on Eq. (11), we shall in the subsequent section, give a critic of 346 the derivation of the Solar radiation balance Eq. (11). 347 8 The emissivity of the Earth shall here be assume to be unity [cf., 74,75]. Taken to the letter, this is obviously not correct because emissivity is defined as the ratio of the energy radiated from a material's surface to that radiated from a blackbody (a perfect emitter) at the same temperature, wavelength and under the same viewing conditions. This ratio varies from 0 to 1, with ( = 1) for a perfect blackbody and ( = 0) for a perfect absorber. The emissivity is dependent on the type of surface and many climate models set the value of the Earth's emissivity to 1. However, a more realistic value is ∼ 0.96 [e.g., 74,75]. gives the name of the planet, its radius, orbital semi-major axis, its albedo, the actual global average temperature T a (t 0 ), the expected global average temperature T pl (t 0 ) in-accordance with Eq. (11), and the last column (7) gives the difference [T a (t 0 ) − T pl (t 0 )] in actual and expected global average temperatures of the listed planets.

348
In the derivation of the RB-Equation (10), we have one major issue, namely that -the 349 fact that, the Earth has an atmosphere, is not taken into account. Our discussion on this 350 perdurable fact will make reference to Fig. (2). It is without any doubt whatsoever that, we 351 have to say: "the fact that, the Earth has an atmosphere -is not taken into account in Earth. The new albedo now simple becomes the ratio of the solid Earth's cross-sectional 366 area to that of its atmosphere, i.e.: In this way, a new model of the Earth's atmosphere is born where the Earth's atmosphere

371
If the Earth's atmosphere is not ignored but taken into account, then, the effective 372 radiation capture radius [R eff (t)] of the Earth system in Eq. (9) will not be equal to the 373 radius of the solid Earth, but, will be equal to the radius of the solid Earth plus the size atmosphere. Therefore, we need to make a correction for this because the atmosphere 377 will certainly capture some radiation. In the next section, we construct a new model 378 based on the criticism here given.  To that end, if -as is the case, the Earth system is a gravitationally bound sys-  Now -we know that: where: G, is the usual Newtonian constant of gravitation. Further, we know that: In-order to calculate: P surf atm (t), we know that the mass, m ⊕ atm (t) of the Earth's atmosphere 406 [see e.g., 78, p.13] is related to the Earth's mean global surface pressure, P surf atm (t), and the 407 Earth mean surface gravitational acceleration, g surf ⊕ (t), by the following formula: where: M SE ⊕ (t), is the mass of the solid Earth at any given time t. Given that: R ⊕ (t 0 ) where, as afore-stated: m * (t) = m ⊕ atm (t)/N atm (t), thus from the condition that at the 415 boundary: F g (t) = F th (t), it follows that: Considering the fact that for all intent and practically purposes: follows that: Out of interest, by making use of the definition of R eff (t) given in Eq. (13), Eq. (19) can 419 be re-written with R ⊕ (t) as follows: In the next section, we shall now link the Earth's albedo to both the supposed expansion 421 rate of the Earth and the changing luminosity of the Sun.

423
In the present section, we shall now act on the criticism that we have levelled against 424 the presently accepted Solar-Earth radiation balance model that is used to derive the RB-

425
Equation (10), where the fact that the Earth has not just an atmosphere, but, a radiation If we are to warm-up to the idea of a gravitationally bound expanding solid Earth whose 437 atmosphere also expands in response to the expansion of the solid Earth, then, according 438 to Eq. (12), the albedo A ⊕ (t) will change too, i.e.: Since according to Eq. (13): R eff (t) = R ⊕ (t) + H ⊕ atm (t), it follows that: From Eq. (19), we also have that: Given that: H ⊕ 0 (t) = k B T ⊕ (t)/4m * (t)g surf ⊕ (t), and assuming steady mean global surface 442 temperatures over the eons for the Earth system, i.e.: δT ⊕ (t) = 0, we will have: where in this Eq. (24), we have made the reasonably good assumption: δM SE ⊕ (t)/M SE ⊕ (t) ∼ 444 0, and that: δm atm (t)/m atm (t) = −(2 + κ)δR ⊕ (t)/R ⊕ (t), and: κ is here some dimen-

445
sionless parameter yet to be determined.

459
From the resultant gravity of the criticism levelled against the RB-Equation (10), there is 460 need for one to take into account the Earth's atmosphere. To that end, if we are to take 461 into account the fact that the Earth has an atmosphere that this atmosphere aids in the 462 10 This assumption that: κ 1, implies that H ⊕ 0 (t) has not changed appreciably over time [i.e.: δH ⊕ 0 (t)/H ⊕ 0 (t) ∼ 0], it has remained constant. This assumption is not unreasonable. We will prove this in the complimentary reading: [81]. capture of some of the incoming Solar radiation, it follows that for the effective power 463 P abs ⊕ (r, t) absorbed by the Earth -instead of it being given by Eq. (6), it will be given 464 by: The difference between Eq. (6) and (34) is the effective radius. We have replaced R ⊕ in 466 Eq. (6) with R eff (t) in Eq. (34).