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,