Geo-engineering Using Mirrors In Space

Let's imagine we can reduce global carbon emissions to net zero: we still need to make the planet colder to stop the positive feedback in the Arctic. There are various geo-engineering solutions that might be deployed to reduce the temperature of our overheating Earth. None of them are easy or cheap but, given the positive feedback we've unleashed, we desperately need to get these technologies working.

One idea (the converse of the Arctic Death Spiral) is to compensate for the reduction in sunlight reflected by Arctic sea ice with man-made mirrors. Some people have suggested placing these mirrors in space in between the Sun and the Earth.

The calculations below estimate the size of mirror required in order to change the Earth's average temperature.

As per A Farewell to Ice p47-49, let's approximate the Earth as a sphere with constant temperature over its whole surface area and use the Stefan-Boltzmann equation for the radiation emitted per unit of surface area of any body. As the Earth is in equilibrium, equate it to the total power the Earth receives from the Sun:
\begin{align} \text{Power emitted as radiation} & = \text{Power received from the Sun} \\ 4{\pi}R^2{\epsilon}{\sigma}{T_0}^4 & = {\pi}R^2S(1-\alpha_{av}) \end{align} where

• \(R\) is the average radius of the Earth, 6.371 × 10⁶m
• \(\epsilon\) is the Earth's emissivity; for a planet with no atmosphere \(\epsilon\) = 1; according to https://en.wikipedia.org/wiki/Climate_model, with our current atmosphere the Earth's emissivity \(\epsilon\) = 0.612
• \(\sigma\) is the Boltzmann constant, 5.67 × 10^-8W/m²/K⁴
• \(T_0\) is the Earth's average temperature, around 15°C or 288K
• \(S\) is the solar constant, 1,370W/m², the amount of radiation reaching the Earth per unit area measured at right angles to the Sun's rays
• \(\alpha_{av}\) is the Earth's (average) albedo, about 0.30

Now let's suppose we place a mirror with area \(A\) between the Sun and the Earth. This area of mirrors will block power from reaching the Earth, as follows:
\begin{align} \Delta_P & = \text{power blocked by space mirrors} - \text{power reflected without mirrors} \\ & = AS - A{P_s}\alpha_{av} \end{align} where

• \(P_s\) is the power of raw sunshine reaching the Earth's surface at midday on a cloudless day, 1,000W/m², according to https://www.withouthotair.com/c6/page_38.shtml - less than \(S\) because some is absorbed by the atmosphere
• NB1: it doesn't matter that some of the Sun's rays will shine round the edges of the mirrors; it doesn't matter which part of the Earth the rays hit; this equation should work if as long as all of the rays blocked would otherwise have hit the Earth's surface
• NB2: this equilibrium overlooks that the mirrors will become warm (because they don't reflect all the Sun's energy) and will radiate some power down to Earth

This simplifies to: $$\Delta_P = A(S - {P_s}\alpha_{av})$$

Let's insert this delta into our initial equilibrium equation:
\begin{align} \text{Power emitted as radiation at temperature } T_m & = \text{Power received from the Sun} - \text{incoming power blocked} \\ 4{\pi}R^2{\epsilon}{\sigma}{T_m}^4 & = {\pi}R^2S(1-\alpha_{av}) - A(S - {P_s}\alpha_{av}) \end{align} where

• \(T_m\) is the new (lower) equilibrium, average temperature of the Earth with the mirrors deployed

We can substitute the right side of the equation from our initial equilibrium equation: $$4{\pi}R^2{\epsilon}{\sigma}{T_m}^4 = 4{\pi}R^2{\epsilon}{\sigma}{T_0}^4 - A(S - {P_s}\alpha_{av})$$

Rearranging we get $$A = \frac{4{\pi}R^2{\epsilon}{\sigma}({T_0}^4 - {T_m}^4)}{S - {P_s}\alpha_{av}}$$

Suppose we want to reduce the temperature of the Earth by 1°C, ie \(T_m\) = 287K then \begin{align} A & = 1.57 \times 10^{12}m^2 \\ & = 1,570,000km^2 \end{align}

That's a square of mirrors measuring 1,250km on each side, which needs to be built in space. That's a massive project! Let's hope we find more feasible geo-engineering solutions.

https://en.wikipedia.org/wiki/Space_mirror_(climate_engineering) suggests that

the deployment and maintenance of a fleet of small space mirrors that can create a shade of around 100,000 km² in space would include necessary factors such as energy, construction, transportation, and ground support operations. Overall, the estimated cost of constructing and sending a fleet of space mirrors to space is around 750 billion dollars. If the space mirrors are able to achieve a 50-year lifetime, the annual maintenance cost estimates to around 100 billion dollars. Furthermore, if any individual satellite needed to be replaced at the end of their lifetime, the costs of the entire operation would come to around 5 trillion dollars.

The calculation above suggests that the mirror needs to be 12 times larger, ie costing 60 trillion dollars. Expensive as this idea might be, the alternative of not geo-engineering will lead eventually to even worse outcomes.

See also Geo-engineering Using Terrestrial Mirrors and Geo-engineering By Reforesting