Building Planets: Explaining Expression (??) in Planet Construction

[vaibhavtewari]

Hello Everyone,
I kind of followed this document

http://pages.physics.cornell.edu/~aalemi/random/planet.pdf

until on page 3, author explains

What we are trying to due, is build the planet such that each chunk con-
tributes as much as possible to this integrand. So, if we pause for a second
and think about expression (??) as a sort of measure of cost effectiveness, it
isn’t long before we realize that our planet’s surface should correspond to a
constant contour of this expression.

which does not makes too much sense to me. Can some one explain this to me, or give an alternative solution.

Thank You

 

[A.T.]

Hello Everyone,
I kind of followed this document

http://pages.physics.cornell.edu/~aalemi/random/planet.pdf

until on page 3, author explains

What we are trying to due, is build the planet such that each chunk con-
tributes as much as possible to this integrand. So, if we pause for a second
and think about expression (??) as a sort of measure of cost effectiveness, it
isn’t long before we realize that our planet’s surface should correspond to a
constant contour of this expression.

which does not makes too much sense to me. Can some one explain this to me, or give an alternative solution.

Thank You

"constant contour of this expression" means a surface where that expression ( equation (1) given above the quoted paragraph) is the same at every point.

To maximize the integral that gives you g at some point, while minimizing the used mass, you have to fill the space with mass in the order in which each point contributes to the integral (most contributing points first). So the planet surface will be an iso-surface of the spatial function that gives you the contribution factor for each point.

 

[vaibhavtewari]

thanks for replying but this is almost reworded version of the statement...can you show them in equations and math

To maximize the integral that gives you g at some point, while minimizing the used mass, you have to fill the space with mass in the order in which each point contributes to the integral (most contributing points first).

Thanks

 

[Bill_K]

vaibhavtewari, If the surface of the planet did not follow the contour, there would have to be some of the planetary surface sticking outside the contour and some of the planetary surface still inside the contour. And then if you took some of the part that's outside and moved it to a place where the surface is inside, you would increase the gravity. So by contradiction, when you can no longer do this, none of the surface remains outside.

 

[A.T.]

What Bill said.

A very nice problem BTW, with a rather disappointing result: "So, after all that work, in the end of the day, you can only do about 1.03 times better than the sphere if you want to maximize your gravity"

 

[vaibhavtewari]

can you show it in math. Like this is the integral, something that is not worded. I am pretty sure any thing in physics that can be worded can be written in equations.

Thanks