Exploring Gravity Wells: The Sun's "Awfully Long Way Down

[Dr Wu]

I recently came upon a wall chart that shows the 'gravity wells' of the various planets in our solar system. the chart does include a 'noises off'- style mention of the Sun's own gravity well, which is appended by the unhelpful comment: 'An awfully long way down'. I've since found out that for the Sun it's around the 19,000,000 km mark - which compared to the Earth's own 6,400 km gravity well, is indeed an awfully long way down.

It did get me thinking, however. I assume the depth of a gravity well is determined purely by the mass of a given object, rather than its density or surface gravity. As an example: were the Sun to be transformed into a white dwarf or even a black hole, would its gravity well be the same, I wonder? I should expect so, but I cannot be entirely sure. Indeed, I'm prepared to be proven wrong - especially when it comes to black holes, which seem to defy commonsense at every turn.

 

[Bandersnatch]

were the Sun to be transformed into a white dwarf or even a black hole, would its gravity well be the same

It would be the same down to the point marking the old surface (before contraction), so everything above that point would behave identically in both cases.
It would be different below the surface point, because, as you said, the amount of mass pulling you down is what matters,, and it is different in these two cases (i.e. with the regular star as you go down below its surface, you're leaving some mass behind; whereas with a compacted star you still have all that mass below).

 

[Dr Wu]

Thanks, Bandersnatch. So, noting your qualifying remark regarding the surface point, I take it then that a solar-mass black hole (for example) would indeed have a deeper gravity well? By this I mean an added depth that coincides with the reduced diameter of the hole. I'm sorry if I've not made myself clear here.

 

[Bandersnatch]

I take it then that a solar-mass black hole (for example) would indeed have a deeper gravity well? By this I mean an added depth that coincides with the reduced diameter of the hole.

Yes.
If you look at the infographic you're talking about, which I'm pretty sure is this one:
https://xkcd.com/681/
they provide the formula they used for scaling the picture, where depth of the well is proportional to mass and inversely proportional to radius. If you reduce radius and keep the mass constant, you get deeper well.
With black holes it either means that you get a much deeper well (if using event horizon as the radius), or an infinitely deep well (if using the infinitely small singularity).

 

[Dr Wu]

Well, I did the calculations based on the formula contained in the chart, and after a few initial wild errors, I managed to derive the Sun's gravity well to about 19,400,000 km - which for me is pretty darned good. I also did the same for the Earth, and that checked out fine too. Then purely for the hell of it, I shrank the Sun down to a 20 km diameter neutron star (while retaining its original mass) and came up with a gravity well that measured a jaw-dropping 1.4 trillion kms! Surely that can't be right? Or if it comes to even within shouting distance of being correct, then I'm fully prepared to be amazed.

 

[Bandersnatch]

1.4 trillion km

Looks about right.

 

[Dr Wu]

I'm amazed.

PS. That Isaac Newton was a clever fellow.