TImeline of Rogue Earth

Question

I have seen a video from Kurzgesagt that goes into some details about what would happen should the Earth be flung outside the solar system. It says that by the time Earth's altitude relative to the sun would be as high as the average orbit of some planets, the surface temperatures would be around some ranges.

Then I found this question: The earth is flung into deep space

For which jdunlop has this answer:

There's actually a popsci article about exactly this.

Pertinent:

Within a week, the average global surface temperature would drop below 0°F. In a year, it would dip to –100°. The top layers of the oceans would freeze over, but in an apocalyptic irony, that ice would insulate the deep water below and prevent the oceans from freezing solid for hundreds of thousands of years. Millions of years after that, our planet would reach a stable –400°, the temperature at which the heat radiating from the planet's core would equal the heat that the Earth radiates into space, explains David Stevenson, a professor of planetary science at the California Institute of Technology.

However, neither the video nor the article present an actual timeline, i.e.:

Day 0: nothing is really much different, no temperature perceived
Day 50: average global temperature falls by X degrees; Earth crosses Mar's orbit.
Day 100: average global temperature is Y, polar caps extend all the way to latitude 45, Earth crosses Jupiter's orbit

And so on.

I understand that there is an infinitude of angles that a massive body could approach the Earth in order to fling it into space; that some trajectories might bring us closer to the sun before we are finally ejected; and that for different trajectories, we will have different timelines. What I am looking for is one specific scenario, with the following hard requirements:

1. The Earth is only ever accelerated prograde - this will keep it from approaching the sun, while also keeping things simple(r) (I hope).
2. The Earth is accelerated from its current 30km/s to 45 km/s orbital speed just once. No multiple passes.
3. The acceleration is gentle enough that humanity can survive it. We're all going to die for a number of reasons as the Earth cools down, but the initial acceleration itself should be survivable by at least some people.

And one optional requirement:

1. The Earth keeps the Moon orbiting around it.

I am giving Earth a boost to about its escape velocity + approximately 3 km/s in this case, so from that I could work on figuring out when the Earth would cross each planet's orbits - but I don't know how to figure the temperature changes by week, month or year. Please help me with that. Notice that the video and the article in the links provide no initial departing speed.

For the purpose of this question, and to simplify things further, we could consider that all planets have 0 eccentricity orbits. What I would like is a table with most probable dates for some milestones, i.e.:

Days from Acceleration event | Average Global Temperature
---------------------------------------------------------
0                            | 1.16 C
X                            | -10 C
Y                            | -20 C
...                          | ...
Z                            | -50 C
And so                       | and so

I'm okay with temperatures in F or K too.

500 rep bounties for probable times when:

• The polar caps meet
• The atmosphere freezes
• The orbits of planets are crossed

Answer 1

I remembered that Kurzgesagt always lists sources for their material, so I checked and there were some simulations already done by scientists.

Matthew Caplan from the Illinois State University simulated a brown dwarf with about 10% of the Sun's mass passing by Earth's orbit. He provided Kurzgesagt with the following diagrams:

^{Source: the first link in the answer. Click to expand.}

This means that we would pass current Mars's orbit in two months, and Jupiter's in eleven months, and Pluto's average distance from the Sun somewhere around ten and twelve years. The moment when we go past any planet's orbit can be inferred from the center and right diagrams.

As for temperatures, Kurzgesagt's sources point to the following article:

(PDF) The Frozen Earth: Binary Scattering Events and the Fate of the Solar System, Gregory Laughlin and Fred C. Adams, 2000

There are some complex equations there, but from those the temperatures in the video were inferred. The global temperature of the planet would be around -50C at Mars's orbit and -150C at around Jupiter's orbit. A little after Jupiter the polar caps would meet and in ten years (so, close to Pluto's average distance to the Sun) the average ice layer thickness would be ~1 km. Other milestones can be calculated with the equations in the article.