Volcanic eruptions are some of the most powerful calamities on Earth. When they erupt, volcanoes release a large amount of mass in the air just like a jet does.
Does such a blast have any effect on the course of the Earth? Does it change the Earth's orbit in any way? If not, then how big a blast is needed to do it? Would a full eruption of the Yellowstone Supervolcano do it?
I do not think volcanic eruptions can change the course of the earth.
This is because whatever materials a volcano ejects ultimately land back to the ground, so the earth remains on its path. So no change in course/radius.
If, however, you can manage to create a blast which sends particles with such a force and such a speed that they never come back again (the speed will have to be $\geq 11.2 km/s$), then we can think about the change of course of the earth. Can Yellowstone do it?
But even if that DOES happen, the mass which it will be able to eject completely out of earth's gravitational field will be very less compared to the mass of the earth. So we can neglect and tiny change, since it's almost nothing. However, if a volcano could blast off even $1/10^{th}$s of the earth's mass, then we'd have some change in course/radius.
Addendum: I just thought of one thing though-
The material thrown up in a volcanic eruption emerges from inside the earth. When the eruption is over and the material is back on the ground, there has been a slight net displacement of the materials. What was first a few kilometres under the surface is now lying on the surface. Therefore, there may as well be a tiny change in course.
HOWEVER, as pointed out before, the mass thrown up by a volcano is very very very less compared to the mass of the earth and the displacement of the material is very very less compared to the radius of the earth ($\approx 6400 km$), we can safely neglect this tiny change in course.
Even if there is a change, the radius of the earth's orbit around the sun is 150,000,000 km. A change in course by a fraction of a centimetre will make NO difference at all.
Addendum-2: Akash points out that the materials striking the ground don't apply the same force as the force at which the volcano erupts.
That is not true. It may seem that the force is less, but that is because the materials are fragmented and a lot of the force is applied onto air too.
The force is distributed between all the particles and the air. When each particle strikes, it imparts its force to the earth. The air 'falls' back too, applying a force too. If you add up the forces, it will be equal to the force of the volcano.
Unless, of course, the materials are propelled out of the earth's gravitational field.
The effect would be less than that of the weather, or even large earthquakes.
By “earth’s course” you could mean earth’s orbit around the sun, the earth’s rate of rotation about its own axis, and the orientation of the earth’s axis relative to distant stars.
Any process that involves the redistribution of mass can affect the motion of the earth – an earthquake, a volcanic eruption, seasonal weather, ocean currents, or driving your car. Although the effect can be calculated, the redistribution of mass by driving your car would much too small to measure using present technology. The most advanced methods of measuring earth orientation were required to detect that the great 2011 Japan Quake shortened the length-of-day and moved the earth’s axis. The effect of a volcanic eruption has not yet been measured. Because a volcanic eruption would redistribute relatively less mass, the calculated effect of the event upon the oblateness and moment of inertia of the earth would be less than an earthquake and much less than seasonal weather.
The formation of a shield volcano the size of the Olympus Mons (on Mars) on Earth would change the Earth’s rotation from what it is presently. The distribution of mass associated with the Yellowstone hotspot contributes to the Earth’s present moment of inertia.
Shoot out a pea attached with a spring to the big lead ball. Nothing will change for the course of earth as the (minimal shifted) centre of mass of the whole system keeps it initial course it always had.
If the force is so great that the spring snaps and the pea does get detached, the c.m. wouldn't change either as the recoil moment would push it back. Earth will have a wobble for a while, being dampened by releasing heat.
Entropy is rising so thermodynamically sound.
