Bdtyktl

Without worrying about the other issues of global warming like rise in CO$_2$ level, hole in ozone..etc, lets assume that we're facing with rise of global temperature. And assume there's an efficient way to make earth's orbit wider or shorter around the sun.

Now, would it solve the problem if we move the earth farther away from sun, just enough to keep the global temperature constant, as it gets heated by global warming?

Beware of unintended consequences!

You would have to move it quite a long way. Remember that the difference between winter and summer is due to the tilt of the Earth's axis more than its proximity to the Sun.

Doing so would change the length of a year and could disrupt a lot of living organisms' life-cycles.

For years biologists have wondered why cicadas emerge from their
underground habitats after a prime number (7, 13 or 17) of years.

http://www.abc.net.au/science/articles/2001/11/27/421251.htm

Moving the moon at the same time and still keeping it in orbit would be tricky as well. We need the Moon for the tides and for the reproduction of many creatures. Actually the differentiation between spring and neap tides would diminish because the Sun would have less gravitational effect.

A spring tide—popularly known as a "King Tide"—refers to the
'springing forth' of the tide during new and full moon. A neap
tide—seven days after a spring tide—refers to a period of moderate
tides when the sun and moon are at right angles to each other.

https://oceanservice.noaa.gov/facts/springtide.html

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How Moonlight Sets Nature’s Rhythms

https://www.smithsonianmag.com/science-nature/how-moonlight-sets-nature-rhythms-180963778/

Another problem is that the next ice-age might be hastened enormously. Our existing system is not stable in the long-term even when we exclude the excesses of humanity.

There are many factors to take into account and hence the high likelihood of unintended consequences. Even the best simulation would likely miss some of these and the movement would have to be calculated carefully and subject to numerous periodic adjustments until things settled down.

Milankovitch cycles describe the collective effects of changes in the
Earth's movements on its climate over thousands of years... variations
in eccentricity, axial tilt, and precession of the Earth's orbit
resulted in cyclical variation in the solar radiation reaching the
Earth, and that this orbital forcing strongly influenced climatic
patterns on Earth.

https://en.wikipedia.org/wiki/Milankovitch_cycles

Clearly if you had the capability to move a planet without actually destroying its surface, something as trivial as fixing its atmospheric chemistry would be no problem at all.

And assume there's an efficient way to make earth's orbit wider are shorter around the sun

No matter how efficient it is, there are certain energy requirements that can't be avoided.

The mass of the Earth is about $6times 10^{24},kg$ whereas the mass of Earth's tiny atmosphere is just $5times 10^{18},kg$. That's less than a millionth the mass of Earth.

And this of course ignores the complication of moving the Moon with Earth without catastrophic consequences for Earth, as mentioned by "chasly from UK".

The effective temperature formula can be used to make a rough estimate of the change in orbital radius required to make a one degree change in temperature. The formula gets us :

$$frac {T_1} { T_2} = sqrt { frac {R_2}{R_1} } $$

or :

$$R_2 = R_1 left( frac {T_1} {T_2} right)^2$$

So a change of $1^text{o} K$ from $T_1=252^text{o}K$ (Earth's effective temperature ignoring the effect of atmosphere) to $T_2=251^text{o}K$ gives us a radius $R_2=1.008AU$. So a small change of "just" eight thousandths of an AU.

So how much energy is required to move the orbit that much ? Well we'll approximate that with the change in kinetic energy of Earth (ignoring the Moon).

Orbital speed is, to a good approximation :

$$v = sqrt {frac {GM_s} R}$$

where $M_s$ is the Sun's mass.

Kinetic energy is given by :

$$E=frac 1 2 m_e v^2 = frac 1 2 m_e frac {GM_s} R$$

So the change in kinetic energy is :

$$Delta E = frac 1 2 GM_sm_e left( frac 1 {R_1} - frac 1 {R_2} right) approx 2.1times 10^{31}J$$

The energy required to remove Earth's atmosphere and replace it entirely with a new one would be of the order of twice the potential energy of the atmosphere. A rough estimate of that is :

$$Delta E_A approx 2frac {GM_Am_e} {r_e} approx 6.3times 10^{26} J$$

So (not surprisingly) moving the old atmosphere out and moving the new one in takes about three ten thousandths of the energy that changing orbit does.

And of course you don't really have to completely replace the entire atmosphere, just do what, on this scale, is a little fine tuning to the chemistry.

Depending which specific consequences of warming you don't like, it may be cheaper and more feasible to move the Moon.

Let's say sea-level rise is the main danger you want to mitigate: The trouble with that in most parts of the world isn't the rise in mean sea level, it's the higher high-tide peaks. Moving the Moon out would reduce the magnitude of its tidal effect, as well as the period length. Sea level could rise by several feet but tidal peak heights could still be reduced.

As others mentioned, now you've got a whole world's worth of side effects and potential unintended consequences to consider. Is there any plausible way to imagine tidal reduction as a driver of additional effects which would have the tendency to cool rather than warm the globe? Biosphere impacts would be severe - it's easy to imagine the harm but maybe it would enable some mechanism which could result in the level of carbon-sinking needed to mitigate the warming and possibly introduce a powerful homeostatic regulating effect. Trading off massive species loss and biosphere disruption today for many, many millenia of engineered equilibrium.

I think so. Sadly, I am not able to deliver any good equations for that but as it would be colder further away from the sun, it would certainly be possible to cool down the average temperature. However, there are some other complications:
The solar system is a chaotic environment that is naturally unstable. If you zipped the earth from one place to another in one moment, this might change the orbits of some asteroids etc, increasing the risk for earth to be hit by such. Of course, this risk would still be fairly low.