Talk:Gravitational binding energy

A gravitationally bound system has a lower (i.e., more negative) gravitational potential energy than the sum of its parts
"the sum of its parts" is vague and ill-defined, the way it is phrased here.

Clarification required

 * It is equal to minus the gravitational potential energy, but should not be confused with the case of separating e.g. a celestial body and a satellite to infinite distance, keeping each intact.

I have to confess that this sentence only served to support exactly the confusion it warns against. Can anyone clarify? Is gravitational binding energy the energy required to blow an object to smithereens, all infinitely separated? -- pde 01:40, 12 Mar 2005 (UTC)


 * Yes, it's the energy required to break up the object to infinitely small fragments and separate them infinitely. 193.171.121.30 21:51, 18 December 2005 (UTC)


 * This subject came up in a discussion of the Nordtvedt effect and the reply I got was that Gravitational binding energy was negative. The details are in this post on sci.astro


 * http://groups.google.co.uk/group/sci.astro/msg/964cd1990593c9d9


 * I wonder if you could clarify this.


 * Thanks
 * George Dishman
 * 83.67.26.177 22:28, 2 January 2006 (UTC)


 * Well, there are different conventions on how the term is used, but as far as I know most times "binding energy" means a positive amount of energy, i. e. the amount of energy which is set free whentheconstituents come together. 193.154.191.21 04:50, 8 January 2006 (UTC) alias 193.171.121.30
 * It's negative compared to zero energy for all the components separated to infinity. Just a different convention. Mordecai-Mark Mac Low 13:45, 26 July 2006 (UTC)

should it be 10% greater than the seperation of two equal mass sphere of 2 r apart ?

Crappy first sentence
The first sentence. "The gravitational binding energy of an object consisting of loose material, held together by gravity alone, is the amount of energy required to pull all of the material apart, to infinity", doesn't make a whole lot of sense. What goes to infinity? There are no units or measurements, is it the energy required? In that case the last comma is useless, if not then the sentence should be better structured so its clearer what the infinity being refered to is. 123.243.215.92 (talk) 11:16, 26 September 2012 (UTC)
 * It means each particle of the object is pulled away so that it is an infinite distance from every other particle (so that their binding energy equals zero). It's strange, but's its just a idealization. 71.104.183.59 (talk) 18:23, 1 March 2013 (UTC)

It's not the only poor sentence. So is "A gravitationally bound system has a lower (i.e., more negative) gravitational potential energy than the sum of its parts". — Preceding unsigned comment added by 2.97.112.134 (talk) 07:13, 20 August 2019 (UTC)

unreasoned removal

 * Would you, Vsmith, please explain why you feel to remove without any explanation the remark:

''Well, doubled mass and a total different mass distribution. But what is the gravitational potential energy, to separate to infinity two half spheres which form a sphere? Then we consider each half-sphere as an object in the gravitational field of the other half-sphere!'' I think at least you should also remove the remark This is 20% greater than the energy required to separate to infinity two such spheres touching each other. Curious of a founded answer. I hope we started not a flame. :-/

I hope everyone agrees about the qualitative improvement by my Ersatz-statement. It is local-topic and not local-off-topic! BTW: I dislike private talks in contrast to open discussion in this Wikipedia -- so everyone can see the arguments respectively sentences of everyone involved. Regards, Achim —Preceding unsigned comment added by Achim1999 (talk • contribs) 14:49, 18 June 2009 (UTC)

Derivation for a uniform sphere
As pointed out by user DVdm in Carl Sagan's Discussion Page "the source really does not qualify for Wikipedia" Deep Atlantic Blue (talk) 22:56, 3 June 2010 (UTC)

Problems with the derivation
I have serious problems with the derivation -- it does not strike me as very rigorous. I've also found with my calculations that the precise result for the Earth is wrong -- it's actually around 1.08 * the uniform-density case. I'll give what I believe to be a better derivation. I'm sure that this derivation can be found in some textbook of celestial mechanics or planetary/stellar structure; does anyone know of any good ones to refer to?

Start with two particles and Newton's law of universal gravitation. Their gravitational binding energy is
 * $$U_{12} = - \frac{G m_1 m_2}{r_{12}} = - m_1 V_2 = -m_2 V_1 $$

for masses m1 and m2, separation r12, and potential V. For several particles, we add up their contributions:
 * $$U = - \sum_{i < j} \frac{G m_i m_j}{r_{ij}} = - \frac12 \sum_{i,j} \frac{G m_i m_j}{r_{ij}} = \frac12 \sum_i m_i V(\mathbf{x}_i) $$

Note the 1/2 that appears; it keeps particle interactions from being counted twice.

Turning sums into integrals gives us
 * $$U = \frac12 \int \rho(\mathbf{x}) V(\mathbf{x}) d^3 x $$

We can turn it into a more convenient equation for the spherically-symmetric case by using vector calculus and Poisson's equation, $$\nabla^2 V = 4\pi G\rho$$, and integrating by parts. Using the acceleration of gravity $$\mathbf{g} = - \mathbf{\nabla} V$$, we get
 * $$U = - \frac{1}{8\pi G} \int |\mathbf{g}|^2 d^3 x$$

In the spherical case, we find this simple form:
 * $$U = - \frac12 \int_0^\infty \frac{GM(r)^2}{r^2} dr$$

It is evident that central concentration increases the binding energy above its uniform-density value.

Lpetrich (talk) 04:38, 1 February 2011 (UTC)
 * Just from taking a look, I can tell that the mathematics in your formula are seriously flawed. — Preceding unsigned comment added by Tonathan100 (talk • contribs) 17:59, 3 October 2015 (UTC)