Talk:Lagrange point

Is the L3 formula correct?

I was working on an assignment and I found this article looking for an analytical approximation for L3 location. The formula in the article was not making sense to me and I checked the reference [18], in which the formula appears in equation (20). The one in the reference does not look like the same and does make sense to me when plugging the numbers. 2001:1C00:B0C:F000:A0B3:5D53:E375:2471 (talk) 00:36, 4 December 2022 (UTC)

Indeed, it was wrong, I fixed it, thanks for noticing. Tercer (talk) 12:24, 20 July 2023 (UTC)

L1, L2, & L3 Quintic Equations

The correct polynomial for the distance between the secondary and L1 is

x^5+(µ-3)x^4+(3-2µ)x^3-(µ)x^2+(2µ)x-µ = 0

and

r = R x

The correct polynomial for the distance between the secondary and L2 is

x^5+(3-µ)x^4+(3-2µ)x^3-(µ)x^2-(2µ)x-µ = 0

— Preceding unsigned comment added by 131.176.243.11 (talk • contribs) 09:03, 19 July 2023 (UTC)

I have verified that both our equations are correct with the MATLAB symbolic toolbox. Here's the details:

r=Rx

Using µ=m2/(m1+m2) where m1 is the mass of the Sun and m2 is the mass of the Earth for the existing force balance equations.

L1: x^5+(µ-3)x^4+(3-2µ)x^3-(µ)x^2+(2µ)x-µ = 0

L2: x^5+(3-µ)x^4+(3-2µ)x^3-(µ)x^2-(2µ)x-µ = 0

As you correctly specified.

Using µ=m1/(m1+m2) (I previously called this expression µ because that's how the existing Wikipedia page was at the time) where m1 is still the mass of the Sun and m2 is the mass of the Earth for the existing force balance equations.

L1: x^5-(2+µ)x^4+(1+2µ)x^3+(µ-1)x^2-2(µ-1)x+µ-1

L2: x^5+(2+µ)x^4+(1+2µ)x^3+(µ-1)x^2+2(µ-1)x+µ-1

As I correctly specified. WaffleJet34 (talk) 18:00, 11 July 2024 (UTC)

L1 Quintic Equation:

This equation is incorrect. To demonstrate, take the arbitrary test case where R = 6; M1 = 10, and M2 = 2, (µ = .8333) the quintic equation has a zero at x=0.8098. Since x = r/R, then r = 6*0.8098 = 4.8586. Plugging these values into the formula above the quintic equation describing the force balance, the left-hand-side becomes 10/(6-4.8586)^2-2/4.8586^2 = 7.5911. The right-hand-side equation becomes (10/(10+2)*6-4.8586)*(10+2)/6^3=0.0079. Clearly these aren't equal so the quintic equation is incorrect.

Using MATLAB's symbolic toolbox, the correct equation was derived to be: x^5-(2+µ)x^4+(1+2µ)x^3+(µ-1)x^2-2(µ-1)x+µ-1. The only difference is the minus sign in front of the coefficient of the x-term. Taking again the previous test case, this equation has a zero at x=0.3414. Since x = r/R, then r = 6*0.3414 = 2.0487. Plugging these values into the formula above the quintic equation describing the force balance, the left-hand-side becomes 10/(6-2.0487)^2-2/2.0487^2 = 0.1640. The right-hand-side equation becomes (10/(10+2)*6-2.0487)*(10+2)/6^3=0.1640. These are equal, verifying the validity of the proposed equation.

L2 Quintic Equation:

Again, this equation is incorrect. First of all, instead of x, it has r as the variable. This results in mixed units throughout the equation as r is not dimensionless. Assuming the r's were supposed to be x's it is still incorrect, however. Taking the same test case as before, the quintic equation has a root at x=0.9025. Since x = r/R, then r = 6*0.9025 = 5.4150. Plugging these values into the formula above the quintic equation describing the force balance, the left-hand-side becomes 10/(6+5.4150)^2+2/5.4150^2 = 0.1450. The right-hand-side equation becomes (10/(10+2)*6+5.4150)*(10+2)/6^3=0.5786. Clearly these aren't equal so the quintic equation is incorrect.

Using MATLAB's symbolic toolbox, the correct equation was derived to be: x^5+(2+µ)x^4+(1+2µ)x^3+(µ-1)x^2+2(µ-1)x+µ-1. This is very similar to the quintic equitation for L1 except all the coefficients are positive. The only difference is the minus sign in front of the coefficient of the x-term. Taking again the previous example, this equation has a zero at x=0.4381. Since x = r/R, then r = 6*0.4381 = 2.6285. Plugging these values into the formula above the quintic equation describing the force balance, the left-hand-side becomes 10/(6+2.6285)^2+2/2.6285^2 = 0.4238. The right-hand-side equation becomes (10/(10+2)*6+2.6285)*(10+2)/6^3=0.4238. These are equal, verifying the validity of the proposed equation.

L3 Quintic Equation

The equation was not written. Once again using the MATLAB symbolic toolbox, the resulting quintic equation is x^5+(µ-8)x^4+(25-6µ)x^3+(13µ-37)x^2+2(13-7µ)x+7(µ-1). Taking again the previous example, this equation has a zero at x=0.0975. Since x = r/R, then r = 6*0.0975 = 0.5850. Plugging these values into the formula describing the force balance in the L3 section, the left-hand-side becomes 10/(6- 0.5850)^2+2/(2*6- 0.5850)^2=0.3564. The right-hand-side equation becomes (2/(10+2)*6+6-0.5850)*(10+2)/6^3=0.3564. These are equal, verifying the validity of the proposed equation. WaffleJet34 (talk) 03:57, 24 May 2023 (UTC)

effects on L3

I have a concern or two about this paragraph …

The L₃ point lies on the line defined by the two large masses, beyond the larger of the two. Within the Sun–Earth system, the L₃ point exists on the opposite side of the Sun, a little outside Earth's orbit and slightly farther from the center of the Sun than Earth is.

Hm, for what mass ratios can one of these two ("outside" and "farther") be true and not the other?

This placement occurs because the Sun is also affected by Earth's gravity and so orbits around the two bodies' barycenter, which is well inside the body of the Sun.

This seems to me much less important than what follows.

An object at Earth's distance from the Sun would have an orbital period of one year if only the Sun's gravity is considered. But an object on the opposite side of the Sun from Earth and directly in line with both "feels" Earth's gravity adding slightly to the Sun's and therefore must orbit a little farther from the barycenter of Earth and Sun in order to have the same 1-year period. It is at the L₃ point that the combined pull of Earth and Sun causes the object to orbit with the same period as Earth, in effect orbiting an Earth+Sun mass with the Earth-Sun barycenter at one focus of its orbit.

How accurate is that last ("in effect…")? Two distinct bodies are not generally equivalent to one body of their combined mass.

One of these days I'll do the necessary algebra, but not now. —Tamfang (talk) 23:42, 15 July 2023 (UTC)

Equation rendering not working

Some equations only appear as LaTeX script. Not being an expert, I cannot see what it missing to permit proper rendering. Ccison (talk) 13:14, 16 August 2024 (UTC)

The equations look fine on my computer. Can you be specific about which ones are not rendering for you? Tercer (talk) 18:48, 16 August 2024 (UTC)

Apologies for responding so late to your question. It now appears that the rendering problem was on my end. After some updates, this problem seems to no longer be a problem. Ccison (talk) 22:33, 7 May 2025 (UTC)

Date formats

@Tercer: Okay, so we're obviously at an impasse on the date formats. I guess I'll be the one to start a proper dialog. We're not seeing eye-to-eye and I don't really want to write an essay explaining what happened yesterday or why. Suffice to say Tercer's original edit summary here was 100% unhelpful, ownership-like behavior and really ticked me off. Always give a valid reason when reverting anything other than vandalism; that's WP:REVEXP. Also, exactly what part of WP:CAUGHTUP don't you understand? I cited that suggestion here and it was again ignored afterwards. I don't get it. May I at least replay my other edits not related to date formats?

Anyway, to the point of date formats, your position is that I shouldn't change the established date format without getting a consensus first on the talk page. I get that and fully agree with that point, and you are correct that it is stated in MOS:DATERET right where it says, "Unless there are reasons for changing it based on the topic's strong ties to a particular English-speaking country, or consensus on the article's talk page."

What I've pointed out in my edit summaries, and for some reason you're still not getting, is that this article does not consistently apply one date format; it has three. I took the predominant one, as I've done 1000 times, and applied it consistently throughout the article as DATERET says to do in basically all three of its bullet points. So I'm not sure why the resistance to consistency. Is it that you don't like MDY? Because at this point I'll happily go with DMY. But let's please choose a date format and apply it consistently throughout the article. This is too trivial to waste much more time on. — voidxor 14:14, 7 May 2025 (UTC)

Fair enough, I have partially restored your WP:CAUGHTUP edit. I didn't think it was worth preserving, since it was mostly automated non-visible changes. The substantial part was deleting a true but unsourced statement. Instead of eliminating it I added a source.

As for the date format, the problem is that there isn't a predominant one. If anything yyyy-mm-dd is the most commonly used, and that's what you should use if you absolutely need to standardize it. But I really don't think you should standardize it.

I do apologize for my initial unhelpful edit summary. I got angry because I really don't want to waste any time arguing about date formats, and I just wanted it to go away. Of course, the effect was the exact opposite. Tercer (talk) 16:05, 7 May 2025 (UTC)

Thank you for your comments above; they make me feel a lot better! Thanks as well for finding and adding a reference! My only other thoughts are twofold:

• You keep referring to DATERET, which emphasizes consistency. MOS:DATEUNIFY goes into more detail. If fixing it on this article is going to cause stress right now, it can wait.
• yyyy-mm-dd, which comes from the ISO 8601 standard, is not acceptable in prose. Per MOS:DATE, it can only be used "in limited situations where brevity is helpful". That includes tables and maybe citations, though I would argue DATEUNIFY should prevail and we should use the same format in prose and citations. That's why articles can be tagged as using DMY or MDY dates but not YMD. Likewise, the script I'm using doesn't support YMD. — voidxor 16:29, 7 May 2025 (UTC)

I'm glad we could have come to a civil understanding. If the issue is prose versus citations, though, the solution is simple: there is a single date in the entire article, and it is in the DMY format. It is therefore trivially consistent, and there's nothing to fix.

MOS:DATEUNIFY does not prescribe that the dates in prose and the citations must have the same format; it is explicit that they might be different. I think you have three options then (1) Leave them as they are, (2) Standardize to the predominant style yyyy-mm-dd, (3) Change to the prose style DMY. Tercer (talk) 16:51, 7 May 2025 (UTC)

Okay. Yesterday I'd counted overall; I didn't break it out by prose versus citations. YMD should be avoided in prose. With the script I can easily switch the whole article to DMY, if that's cool with you. — voidxor 17:31, 7 May 2025 (UTC)

I won't revert you, if that's what you're asking. Tercer (talk) 17:49, 7 May 2025 (UTC)

 Done, thanks. — voidxor 19:10, 7 May 2025 (UTC)