Okay, guys. Interesting physics question that I made up

Okay, guys. Interesting physics question that I made up.
I don't know the solution yet. Consider the following situation. If you want, please submit anything that you want, but don't be idiots. Maybe anything arithmetic, too? Stuff you'd see in highschool textbooks, but problems you'd actually like to solve out.

Here's my problem:

VY Canis Maioris is a very large Hypergiant, and is one of the largest known stars. Pretend it exists a few light years away, and its relative velocity to our solar system is ~+20,000 meters/second initially. Because of the scale, assume that it doesn't matter that they're in angular motion around Sagittarius A*. Assuming that it's approaching the solar system at an angle of 3π/2 rads, assuming that the right hand position is 0 rads or 2π rads.

The solution to this problem is, "At what time will the pull of VY Canis Maioris over power the sun's gravitational pull on the Earth, and cause the Earth to orbit it instead?"

Easy mode: Assume the Earth is remaining still at this time, and for these purposes, it has ZERO rotational velocity; and everything in the solar system doesn't move either.

Normal mode: Assume the Earth has a constant orbital velocity. These will be in the givens below.

Hard Mode: Assume that all the planets start in the same position as well. When will THEY ALL begin to orbit VY CM?

Pretty damn hard mode: Now, factor in the angular movement of them. They are the same as their linear movement.

Note: there are other factors that might affect this like solar wind and shit. You can discount all of it.

Constants/info:
(Law of universal gravitation)
F=(G*m1*m2)/r^2
(Gravitational constant)
G=6.674e-11 nm^2/kg^2
Mass of the Sun: 1.989E+30 Kg
Mass of the earth: 5.92E+24 Kg
Mass of VY Canis Maioris: ~5.96E+31 Kg
Earth's mean distance: 149,597,870,700 meters (1.5E+11 m)
Velocity of Sun: 220,000 m/s (Relative, zero.)
Velocity of VY Canis Maioris: 420,000 m/s. (Relative, 200,000 m/s)
The velocity of Earth: 460 m/s.
Approach angle of VY Canis Maioris: 3π/2 (270°)
θ1 of the Earth: also 3π/2
Mars' v:2.41e+4 m/s
Mars' Mass: 6.42+e23 kg
Mars' mean distance:2.279+e11 m
Jupiter's v: 1.31e+4 m/s
Jupiter's mass:1.90+e27 kg
Jupiter's mean distance: 7.786+e11
Saturn's v: 9.6+e3 m/s
Saturn's mass:5.69+e26 kg
Saturn's mean distance: 1.433+e12 meters
Uranus' v: 6.8e+3 m/s
Uranus' mass:8.68+e25
Uranus' mean distance:2.873e+12 meters
Neptune's v: 5.4+e3 meters/s
Neptune's mass:1.03+e26 kg
Neptune's mean distance:4.495e+12 meters
Mercury's v: 4.74e4 m/s
Mercury's mass:3.3e+23 kg
Mercury's mean distance: 5.79e+10 meters
Venus' v: 3.5e+4 m/s
Venus' mass: 4.87e+24 kg
Venus' mean distance: 1.082e+11 meters
>disregard eccentricity and assume that there is no y component; assume it's 2-d.

Really fuckin difficult mode: Include planetary interaction in this. Oh, god.

Distance to Canis Majoris or gtfo.

Isn't Canis Majoris bigger than most of the solar system, like to the orbit of Jupiter or Saturn? Would Earth even start orbiting the star or just get pulled in and swallowed. Or maybe just flung off into space if we were lucky.

I'm pretty sure I gave that? It's one light year, or 9.4607+e15 meters. This is not an actual estimation, it's only for our purposes.

You said "a few", so that's why I was confused.

My bad, dude. Just use a light year, aka that distance. Damn, I could've sworn i said that.

Also, at that scale, the difference in time for each planet to go into orbit would be insignificant.
Not only that, but the timescales we're dealing with are so enormous that it would be on the order of 10^10 seconds, or about 1000 years. At that point, orbital perturbations would be so severe that it's almost meaningless to approximate.

sounds like a valid answer. Should we change the problem to make it suitable? Like change the star or distance, maybe?