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Suppose that due to an encounter with a rogue planet, the Earth is propelled out of the Solar system. It will then get gradually colder on Earth. We can try to restore the global temperature to the old level by injecting greenhouse gases into the atmosphere. There are greenhouse gases that are more powerful than CO2, e.g. methane is about 80 times as powerful than CO2 but has a short atmospheric lifetime of about 12 years, liming the concentration in the atmosphere we can achieve. But HFC-23 is about 12,000 times as powerful a greenhouse gas compared to CO2 and has an atmospheric lifetime of about 200 years. Still, the greenhouse effect behave logarithmically on the greenhouse gas concentrations, so at larger concentrations the incremental effect will go down per unit greenhouse gas added to the atmosphere.

Suppose then that as the Earth moves farther and farther away from the Sun, we increase the concentration of various greenhouse gasses more and more to keep the ambient temperature the same. What is the maximum distance beyond which this won't work anymore?

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    $\begingroup$ If Earth is knocked into a cometary or escape orbit, it'll get too far from the sun too quickly for atmosphere modification to do any good. It won't be a slow process over many years. I also seem to recall a similar question being asked before, but I can find it right now. $\endgroup$ Mar 30 at 15:15

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Plants require some amount of light. They don't need current level, but below 25% of current level they will start to decrease production. So x2 distance is the limit for agriculture as usual.

100 ppm increase in CO2 changed heat balance by about 1000 ppm.

https://www.nasa.gov/vision/earth/environment/earth_energy.html

Assuming we can do 1% atmosphere change at most, this will give 10% heat balance change for co2. Distance is 5% more.

For high GWP gas, with a value of 10 000 times the CO2 this change would be 10 000 times greater. Heat balance 1000% change. Distance change 300%. Or distance x4. This is further than what plants will not notice, but this is within a range where most plants still can grow. This location is the asteroid belt, between Mars and Jupiter.

Keep in mind, single event will only move one half of the orbit. The other half of the orbit will stay where it was. So if you will make the planet optimal for the far side of the orbit at 4 AU, earth will overheat during the 1 AU part of the orbit.

If we account this, we could assume that planet might be somewhere at 8 AU, closer to Saturn, with half of its way being overheated and half of it being frozen. But further part of the orbit is significantly slower than the inner part. I don't know how to calculate it. I assume for 10 000 GWP, 1% gas in atmosphere optimal distance of the far side goes to about 5 AU rather than 4 AU, and that's it. It is Jupiter's orbit. Also keep in mind the year will be many times longer.

If we make it extreme, at 30% atmosphere, far side will be at about 30 AU, Neptune. Year will be closer to a current 100 years, so pretty much no matter what you do, everything dies. Twice, from overheating while close and from freezing, including air and this gas, in the far side.

If some magic just so happen to arrange a second rouge planet to meet Earth on the far side, and orbit becomes circular again, then for 10 000 GWP, 30% atmosphere optimal distance is 20 AU, with year being 80 years. Sun's light intensity will drop to just 0.2% of what it used to be. Pretty much all plants will die from not enough light, and animals from having not enough food.

Major problem with your idea is that almost all of high GWP gasses are not stable at low temperature. All of them will just fall as a rain or snow in polar region. CO2 is the only high GWP gas that is stable in polar region. So none of what I've written above is important anyway...

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