The mass of the Earth is: 6.0 * 10^24 kg (
Here)
We now use the formula from
Here along with the definition of
the Semi-Major Axis.
The Semi-major axis in this case, is 1 AU = 1.496×10^11 m.
Thus E<Mech> = -G * M * m / 2a = (-6.67300 × 10^-11 * 1.9891 ×10^30 * 6.0 * 10^24) / (2 * 1.496×10^11)
= 26.62 * 10 ^ 32J
And finally, The total energy = U + K
= 2E/(1-e) - (E(1+e)/(1-e))
= (2E - E - eE)/(1-e)
= E(2 - 1 - e)/(1-e)
= E(1-e)/(1-e)
= E
This makes the total energy in the orbit of Earth 26.62 * 10 ^ 32 J
So, assuming that all the nukes occur on places at midnight, then the amount of energy needed to make the Earth go into the sun is 2,662,000,000,000,000,000,000,000,000,000,000 Joules.
A 1 kiloton nuclear warhead produces 4.19x10^12 joules of energy (
Here) This means that you would need a 6.35 * 10^20 KiloTon Nuclear weapon to do this... Americas largest nuclear weapon is the MK-41 Nuclear Weapon, with a payload of 25 MT (25 * 10^3 Kiloton), meaning you would need 25,400,000,000,000,000 of these nukes)
Now... assume you PLAN to send the Earth into the Sun... thus you could launch a nuke or two every year, at the same time, at midnight, and make the Earth go into a crazy eliptical orbit.
The only requirement NOW is that the orbit is an ellipse, where the distance from one focus, to a point is 1 AU, and the distance from the same focus, to the other side of the ellipse is 1 * 10^9 m (IE: The radius of the sun)
We now use the formula from
Here along with the definition of
the Semi-Major Axis.
The Semi-major axis in this case, is (1 AU + 1 * 10^9) / 2 = (1.496×10^11 + 1 * 10^9) / 2 = 1.506 * 10^11 / 2 = 7.53 * 10^10 m.
Thus E<Mech> = -G * M * m / 2a = (-6.67300 × 10^-11 * 1.9891 ×10^30 * 6.0 * 10^24) / (2 * 7.53 * 10^10)
= 5.29 * 10^33
And finally, The total energy = U + K
= 2E/(1-e) - (E(1+e)/(1-e))
= (2E - E - eE)/(1-e)
= E(2 - 1 - e)/(1-e)
= E(1-e)/(1-e)
= E
So the energy we need to get down to is just 5.29 * 10^33J
Since we started at 26.62 * 10 ^ 32, this means we just need to lose 21.37 * 10^32 J. Using the same logic, that reduces the number of nukes by 5.1 QUADRILLION to 20,300,000,000,000,000