How to Calculate the Lower Limit of the Diameter of an Icy Minor Planet?

[duder1234]

I have a homework question that I am having troubles with.

Q: By equating the pressure at the centre of an icy planetesimal to the maximum pressure that cold ice can sustain without deforming, about 40 MPa, find a lower limit to the diameter of an icy minor planet.

The part I don't understand is the "lower limit to the diameter"
Do I use: P_central >$\frac{GM^{2}}{8πr^{4}}dm$

I just don't know how to get the diameter...

 

[mfb]

I don't know where your formula comes from, but you need some relation between size (like the radius r in your formula?) and pressure in the center, and then let the pressure in the center be 40MPa.
This looks more like an upper limit, however.

 

[duder1234]

I don't know where your formula comes from

I did:
$\frac{dP}{dr}$=$ρg$
and $g$=$\frac{GM}{r^2}$
so $\frac{dP}{dr}=\frac{-GMρ}{r^2}$ (Hydrostatic equilibrium equation)
and $\frac{dM}{dr}$=$4πr^{2}ρ$ (equation of mass conservation)

by dividing the two equations: $\frac{dP/dr}{dM/dr}$=$\frac{dP}{dM}$=$\frac{-GM}{4πr^4}$

integration: $P_{c}-P_{s}$=-$\int^{M_{c}}_{M_{s}}$($\frac{GM}{4πr^4}$)$dM$
$P_{c}$ and $P_{s}$ are pressure at centre and surface of the planet
and by setting $M_{c}=0$ and by switching the intergral:

$P_{c}-P_{s}$=$\int^{M_{s}}_{0}$($\frac{GM}{4πr^4}$)$dM$

and

$\int^{M_{s}}_{0}$($\frac{GM}{4πr^4}$)$dM$ > $\int^{M_{s}}_{0}$($\frac{GM}{4πr^{4}_{s}}$)$dM$ = $\frac{GM^{2}_{s}}{8πr^{4}_{s}}$

hence

$P_{c}-P_{s}$>$\frac{GM^{2}_{s}}{8πr^{4}_{s}}$

and by approximating that the pressure at the surface to be zero ($P_{s}=0$)
we get:

$P_{c}$>$\frac{GM^{2}_{s}}{8πr^{4}_{s}}$

you need some relation between size (like the radius r in your formula?) and pressure in the center, and then let the pressure in the center be 40MPa.

So are you saying I should do:

$40MPa$>$\frac{GM^{2}_{s}}{8πr^{4}_{s}}$?

I figure I have to solve for r and then obtain the diameter from there but I am stuck because I am not given the mass... (or do I use a mass of an icy minor planet like Ceres?)

 

[mfb]

Your formula differs from the one in the first post now.

You know the density of ice, this gives the relation radius<->mass,

 

[pmacias]

A: The "lower limit to the diameter" refers to the smallest possible diameter that an icy minor planet can have while still maintaining its shape and not deforming under the maximum pressure of 40 MPa. In order to find this lower limit, you will need to use the equation you mentioned, Pcentral >\frac{GM^{2}}{8πr^{4}}dm, which relates the central pressure (Pcentral) to the mass (M) and radius (r) of the planet. By equating this pressure to the maximum pressure of 40 MPa, you can solve for the minimum radius (and therefore diameter) that the planet can have. This will give you the lower limit to the diameter of an icy minor planet.