

copy of same material in Climate Control under Global Warming COOLING OR HEATING A PLANET BY USE OF RINGS J.H.L.Lawler Ó 1984 IJS Vol 4 page 33
It is possible either to cool or to heat a planet, or to redistribute the heat, cooling the equator while warming the poles, by use of rings of reflective material about the planet. For example if a planet were to be too hot at the equator, that could be climatically altered by placing a fairly low orbit ring to block and reflect part of the isolation that would have struck the equator, thus radiating' it to space. and reducing the total heat input, In the reverse case a ring placed so that it does not obstruct incoming light, but so that it does reflect some light from the ring to the planet that would not have Otherwise reached the planet can raise the temperature by rather large amounts, up to double being easily obtained.
In the case of a low altitude ring the solid angle Subtended by the planet and by the open space are approximately equal, and thus half the reflected light would reach the planet, and half would he reflected away into space. Thus the heat gathering ability is far less than the heat rejection ability which can approach 95% of the incoming light. The ring can be set to reject selectively into space with virtually none of the reflected light striking the planet, The planet thus is cooled to the extent of the shadowing or the area of the planet times the percent loss from the presence of the ring. In trying to redistribute the energy, The problem becomes more complex. In the most general case there are four distinct areas to be entered into the computation. A_{1 }the area of the planet which is somewhat altered but receives light directly: A_{2 }is the area of the rings which receives light and reflects it to the planet, which would not have directly received light without the rings: A_{3 }is the area which receives light onto the rings which would have normally fallen onto the planet (the shadowed area) and A_{4 }is polar area which is NOT shadowed in part like A_{1} and is not altered at all except by the added liqht from A_{2 } and A_{3}
The total area of the planet is A_{1}+ A_{3 }+A_{4 }as seen from the primary (Sol). The light before the alteration would be LIGHT IN = (A_{1}+ A_{3 }+A_{4}) n_{1} The energy afterward would be: LIGHT IN = A_{1}n_{1 }+A_{2 }n_{2 }+ A_{3}(l n_{2/3} ) + A_{3} n_{3} + A_{4}(n_{1 }+ n_{1/4 })
where the n’s are a combination of the reflectivity and geometry (i.e. the fraction of the planetary surface that receives light from that area) with the third term having been simplified by assumption that the reflected light half strikes the planet and half is rejected to space which is not always true...i.e. that light which passes thru strikes the planet anyhowthat which Is reflected still some strikes the planet. Because of the energy rejection that area Shadowed (A_{1}) is cooled to the extent of the loss by the shadowing area A_{3}, and heated by the added light received from A_{2}note that the cooling is strongest at the equator which receives virtually no A_{2 }light, and tapers to A_{4} which receives maximal benefit with no cooling and much A_{2 } enhancement.
International Journal of Science VOL 4 P 33
Let us examine a point on the equator. It looses light by the ratio of A_{3}R/2 to A_{1 }since the ring A _{3 }reflects, with R, light, but in a low orbit roughly half the light reflected strikes the planet in any case. The light not reflected passes on into the planet thru the ring and is adsorbed to the normal extent of the albedo as if the ring were not there Only the reflected light is lost and Only half of that. However, the energy is increased to the extent of the light reflected from A_{4} (2 parts). Thus depending on the ratio of A_{3} to  A_{2 } the net gain could be positive energy or a net loss resulting in negative energy balance. If the ring crosses at 30º North and 30° South latitude (note sine 30° = 0.5) then the net energy balance at the equator is zero change. At higher angles the net effect will be cooling, at lower the net effect will be heating of the equatorial regions.  The polar areas (A) have no shadowing at all, and receive 's more light by the ratio of A_{2 } /4 to A_{4}/2 plus A_{3} R/4 +to A_{4}/2 (the pole itself sees only half of A_{2 }and half the light from A_{3} which is reflected (R) strikes the planet if the ring is at low altitude with half the light escaping. At higher altitude a correction for the geometry must also be applied, further reducing the capture from both A_{2 }and A_{3}. The portion of A_{3} away from the poles is fairly compensated by added A_{2 } area added or lost so the net is about uniform at the whole polar area A_{4. }The fractional gain or fractional loss in A_{1 } depends upon latitude, but the effect at any latitude will be proportional to the equatorial effect. The fractional effect then varies in the transitional zone from the equatorial effect to the polar effectlinearly Across the transitional zone. Thus the energy input at any part of the planet, or the net resultant can be computed by simply adding the three zones ( A_{1 }A_{4 }and A_{5}). together. If one wanted to cool the equatorial regions say 1% energy loss and add 2% to the poles then a Specific ring system is designated. It is not possible to quite get uniform global temperature in any real case. A small planet with a huge ring would approach this; however, the ring effect is such that it is possible to moderate any planet~ to raise or lower the mean temperature up to equivalent of about 50% energy either direction, and to shift about half the mean temperature extremes between poles and equator out of the system with a realistic ring system. Any shift past about the point where the ring equals the diameter of the planet in cross measurement would seem te be pushing the limits of practicality,
That however allows areas of 4 times the collection in the ring or double the energy to the planet, so that more than doubles the "habitable zone in planetary systems from that which would be natural. This then allows terraforming planets that require twice the energy or half the energy (the square root of 2 in distance) of the optimal distance, and with some allowance for not using all of the planet as being either too hot or too cold, would allow a zone of about Venus to Mars roughly for acclimatizing planets. Venus is on the marginally close end while Mars is well within the allowable range Mars will be easier, but I have little doubt that Venus may well be terraformed to provide mere living room in the near solar System (lebensraum). These are well within the projected technology of the 21st century and not in the far distant realm of speculation, 
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