Building an Atmosphere for Mars as a part of a Project to Terraform Mars

How do we build an atmosphere for Mars? From the writer of The Terraformers’ Toolkit. (The Terraformers’ Toolkit: Everything you have ever wanted to know about terraforming and our future in space. The where, why, when, how and what of terraforming.) www.amazon.co.uk/dp/B09X6739P...
Mars
Our atmosphere is 78% nitrogen and 21% oxygen, but Mars has just 0.006 bars of carbon dioxide surface pressure. There is more frozen carbon dioxide in the polar icecaps. If this was warmed, it would evaporate into the to make up to 0.2 of a bar atmosphere. This isn’t enough for most life from earth as both plants and animals need oxygen.
Additionally, because Mars has no magnetic field to protect the surface from radiation, a thick atmosphere would be needed.
Carbon dioxide
There is a problem with the amount of carbon dioxide which might enter the atmosphere from the poles on a warmed Mars because humans and many other animals cannot tolerate more than 0.01 bars of carbon dioxide. This is only slightly higher than the amount in the atmosphere today. If all The frozen carbon dioxide entered the atmosphere, there could be 0.2 bars of it which we would not be able to breathe.
This excess carbon dioxide would have to be converted into plant tissue (using the carbon) and oxygen which could be released into the atmosphere. This could be done by plants, but until sufficient oxygen was present in the atmosphere, machines would probably have to do this. Additionally, with Mars’ gravity being only one third of earths’, the atmosphere might have to be three times thicker to achieve the same surface pressure.
Sources
The next issue of where do we get the oxygen and nitrogen from?
Venus
Venus is the closest place to the sun from which we could get these gases. There is certainly plenty of carbon dioxide and nitrogen in Venus’s atmosphere.
Scoop ships to collect it or long tubes could channel it up to a spaceship. However it would take considerable power to get the spaceship away from Venus as the escape velocity is over 10 km/s.
Either approach requires that we separate carbon dioxide into its elemental constituents carbon and oxygen on route to Mars. We would also have to ensure that none of these gases were liberated as the spacecraft crossed earth’s orbit, otherwise we could make our own planet uninhabitable.
Asteroid belt
Bypassing Earth, the Asteroid belt is the next place we might look for these resources.
There isn’t likely to be any frozen oxygen or nitrogen in the asteroid belt, though there is water ice on the outer edge and the largest of the asteroids Ceres is believed to have a sub ice ocean.
We could use the water as a source of oxygen and would have to change the velocity of each asteroid by 3 km/s. This is likely to be cheaper in propellant than getting them from Venus, but it gets cheaper as we get further from the sun.
Jupiter’s Trojans
The planet Jupiter shares its orbit with up to 600,000 trojan asteroids which are larger than 1 km in diameter. Together they’re estimated to have three times the mass of the entire asteroid belt and the mass of volatiles they contain, which is probably mainly water, maybe 5 to 10 times greater.
Gas giant moons
Two of the gas giants’ moons, Titan and Triton, contain supplies of nitrogen. Titan has unique hydrocarbon lakes, seas and rain and will probably be left alone as a place for scientific study.
Triton has a substantial nitrogen atmosphere with some nitrogen frozen on the surface.
Kuiper Belt
Beyond the gas and ice giant planets, we come to the Kuiper Belt. This is a region containing millions of frozen asteroid like objects, orbiting the Sun like a huge doughnut around our solar system.
This is cold enough for volatiles such as carbon dioxide and carbon monoxide, which are gases on earth or Mars, to be found in solid form.
Pushing these objects toward Mars would require fleets of vessels, which could use some of the substance of the asteroids as propellant. The change of velocity needed would be just 0.3 km/s, which is very economical in comparison with venus or the asteroid belt.
However, using this region has a source would mean very long timescales of as long as 10 years or more to get to Mars.
Impacts
Impacting frozen resources into Mars poles would raise temperatures and liberate carbon dioxide. But there is a risk that this might throw more material into space than they add to the planet. Therefore, the majority would need to be aerobraked in the atmosphere rather than impacting the ground.
Summary
We can find nitrogen on Venus, Titan and Triton and we could oxygen on Venus with carbon as carbon dioxide, or with hydrogen in water ice in the asteroid belt, on trojans, on gas giant moons or on Kuiper Belt objects.
We could use solar power to separate carbon and oxygen close to Venus in the inner solar system. Further out we’d need other power sources, probably nuclear, to split water and liberate oxygen.