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This image from Wikipedia of the SE Lagrange points and the Hill Spheres suggests that the SEL points are outside the Earth's Hill Sphere. (The Hill spheres are the circular regions surrounding the two large masses.) Is this correct?

enter image description here

But I thought I've read that both the Hill Sphere and L1 & 2 are 1.5 million kilometers from the Earth.

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    $\begingroup$ Comment on the image: considering that L1 is shown 20% of the distance from the Earth to the Sun, this isn't even close to a representation of the Sun-Earth system, where it's more like 1%. It's just a conceptual schematic of zero-velocity energy contours in the rotating frame of a pair of objects. Unfortunately the image's source says it is and the image tries hard to make it look that way. $\endgroup$ – uhoh Feb 17 at 4:19
  • $\begingroup$ Comment on the question: I think there are other answers in Space SE and here in Astronomy SE that address the inexact nature of the Hill sphere a.k.a. "Roche sphere" (but not Roche limit), so this might or might not end up being a duplicate, but it's nonetheless a good quesiton. $\endgroup$ – uhoh Feb 17 at 4:22
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    $\begingroup$ I wasn't expecting the image to be to scale, but the image does make it seem L2 is clearly beyond the Hill Sphere. I just wanted to be sure L2 is beyond the Hill Sphere and the two were not equidistant from the earth's surface (or center). $\endgroup$ – Bob516 Feb 17 at 4:43
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    $\begingroup$ The short answer is yes, the Hill sphere is defined or bounded by L1 and L2. I'd make this an answer, but my source is wikipedia and if someone with better math and better sources wants to run the formulas, please feel free. I could probably do it, but my math tends to get a little long and ugly. en.wikipedia.org/wiki/Hill_sphere That's the hill sphere calculation. The true region of stability is quite a bit smaller. $\endgroup$ – userLTK Feb 17 at 19:55
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    $\begingroup$ Depends if you regard the Hill sphere as the approximate sphere or the Roche lobe itself. See uhoh's answer here for a graph of the distances to L₁ and L₂ compared with the Hill sphere approximation as a function of mass ratio. $\endgroup$ – antispinwards Feb 18 at 21:09
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I'm going to answer this in a definition-based manner.

Here is the definition of a Hill Sphere according to Wikipedia:

The Hill sphere or Roche sphere of an astronomical body is the region in which it dominates the attraction of satellites.

Here is the definition of a Lagrange point according to Wikipedia:

In celestial mechanics, the Lagrangian points are the points near two large bodies in orbit where a smaller object will maintain its position relative to the large orbiting bodies.

So, if you think about it - L1 and L2 are on the edges of Earth's Hill Sphere - technically, neither inside nor outside, because neither of the two objects dominate gravitational attraction. This is stated on the Wikipedia page for the Hill sphere - and userTLK alluded to this in his comment as well:

In the example to the right, Earth's Hill sphere extends between the Lagrangian points L1 and L2, which lie along the line of centers of the two bodies.

The Lagrange points are where the Sun's and Earth's gravitational influences have no net effect on an object. So, if an object at L1 moved ever so slightly towards the Sun, then the Sun would dominate the attraction of that object, meaning it would be inside the Hill sphere of the Sun. The same goes for the Earth, just in the opposite direction. (The conditions for L2 are different, but detailing them likely isn't relevant to the answer.)

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