Class 16 - Terrestrial Planet Interior Evolution II

In Class 15 we discussed the importance of a planet's thermal history (internal heat driving geological processes) and 2 of the three main heat sources - Formation (accretion) and differentiation (core formation). Next we will add radiogenic heating and make a numerical model of internal temperature evolution of a planet.

To re-cap on HEAT OF FORMATION -

If you take the HEAT OF FORMATION ( see end of Class 15) and divide by the heat capacity of rock, you get the expected increase in temperature associated with this heating - as shown below:

NOTE - these are huge temperatures for the larger TPs. There is no doubt that this sort of energy was generated in forming the planets - the question is this "Did this energy dissipate at about the same rate as it was generated or did it build up quicker than it could be dissipated?" For Earth we think there is still come remnant heat of formation. But for the smaller planets, the heat of formation may well have been dissipated as quickly as the planet formed.

But there is no doubt that the HEAT OF CORE FORMATION was generated quickly and before it could be dissipated. In class we collectively evaluated equation (5.32) in Class 15 for all of the TPs. Here is a re-arrangement of the equation 5.32, separating it into a constant multiplied by 3 factors - so that deltaT = Constant x [1] x [2] x [3].

Note:

1. that we are taking a "typical" value for heat capacity of C~800 Joules kg^-1 K^-1 = 800 N m kg^-1 K^-1
2. that f=mass fraction of core = Mass (core)/Total Mass = rho(core)/rho(total) (Rcore/Rtotal)³
3. apart from the Constant upfront, the rest of the terms are dimensionless.

Below I have make a little spreadsheet of the calculation. Where did I get the core radius for each planet? I looked 8-20, 8-23, 8-24, 8-26 in Hartmann. Since there is some uncertainty in the density of the rock mantle (page 191 of the text mentions a renage of 2800-3900 kg m) I made separate calculations with the low (L) and high (H) values of this mantle density.

Delta T is the estimate of the increase in temperature of the planet due to core formation, in Kelvin.

See the strong dependance on not only total mass but also the size of the core (large temperature increase for Mercury, small for Mars and the Moon).

The third source of heat is RADIOACTIVITY. Below is a table of relevant radioisotopes - also see Figure 8-10.

Compare the magnitude of the heat generation - today - by radioisotopes with the heat generated - now - inside the giant planets Table 8-2 (heat of formation in the case of Jupiter and heat generated by helium rain-out in Saturn) - 2 orders of magnitude larger! Nevertheless, the radioactivity source drives most of planetary geology.

HEAT CAPACITY, CONDUCTIVITY, DIFFUSION

Before we go any farther we need to think about 3 physics terms that we will use in constructing thermal models of the planetary interiors:

1. C or C_p = Heat Capacity -This is the amount of energy needed to heat up a kilogram of material by 1 Kelvin. Above we used a typical value of 800. This is the value for rock. The heat capacity of iron is less - it takes less energy to heat up a block of iron than a block or rock. Ice has a higher heat capacity - it takes lots more energy to heat up a block of ice. NOTE the specific heat capacity here is per kg of material. (PUZZLER on heat capacity - and the answer).
2. K - Conductivity -(WHY can't physics come up with different symbols?? This is DUMB - easily confused with Kelvin) - it is quite intuitive to think of conductivity as how easily or quickly heat is transfered across material - e.g. putting a metal spoon in hot soup vs. putting a wooden spoon in hot soup - which handle can you hold onto for longer? Metal is more conducting than wood. Below you can see that Rock is only a little more conducting than ice but iron is much more conducting than rock.
3. Kappa - Diffusivity - As we shall see below when we start modeling heat transfer in planets, the quantity we really want to use is the thermal diffusivity which is conductivity / (rho x heat capacity) - similar for ice and rock - 10 times greater for iron.

Here's a periodic table of heat capacity for different elements.
Here's a periodic table of heat conductivity for different elements.
Here's a periodic table of density for different elements. Interesting differences - not as correlated as I would expect....

And here's a table I found on the web - showing how much greater the heat capacity of wood is than rock or iron. And then there's the human body.... very difficult to heat up.

SURFACE TEMPERATURES

Before we build our thermal models of the interior, we need to convince ourselves that it's the interior heat that matters - not the heat at the surface coming from the Sun.

1. The amount of heat being radiated from the surface of the Earth very closely matches the solar input. Unlike at J, S, N, the Earth's internal heat flux is small.
2. Furthermore, the solar heat does not penetrate very far down into the Earth - you only need to bury water pipes a meter or so below the surface to prevent them from freezing in winter.
3. So, given the small heat flux from the interior of Earth, how do we calculate the variation of temperature with depth?

SIMPLE MODELS OF THE THERMAL MODELS OF THE INTERIOR

The purpose of the next 2 sections is to show you (a) how heat generated by radioactivity conducts out of the surface and (b) that it is not so hard to construct numerical models.

First, let us consider a uniform planet where the heat generated by radioactivity - Q Joules per kg - is balanced exactly by heat conductivity out of the surface (where heat flux per unit area is K times the temperature gradient - K=thermal conductivity (large for metal spoon, low for wooden spoon - think of stirring hot soup!) What is the STEADY STATE (no change with time) temperature profile?

The temperature at the surface can be estimated by using our familiar equilibrium temperature (page 297) as a function of distance from the Sun and albedo. Plugging in appropriate values, we get the following results, where the different curves correspond to different values of the assumed radioactive heat source.

What happens if we double the size of the planet? You might expect that the total heat generated increases and it is harder for the smaller surface area : volume ratio to get the heat out. Again, the different curves are for different values of the radioactive source Q.

While a smaller planet cools of really easily.

THERMAL EVOLUTION MODELS - i.e. letting the heat generation vary with time (radioisotopes having a half-life) - the planetary interior evolve over time.

We consider a shell of a planet in which the total change in heat of a layer over a unit of time will be the difference between the heat conducting in minus the heat conducting out, plus the heat generated in that time within the layer.

The change in temperature of the layer over the unit of time is the change in energy (delta E) divided by the mass (rho times volume) and divided by the heat capacity of the material in the layer.

Now you may be able to see where the THERMAL DIFFUSIVITY comes from - to calculate the change in temperature (deltaT) over a unit of time (time step deltat) in a shell (of given Area and Volume) we have

This is calculate layer by layer throughout the planet, time-step by time-step.

RESULTS! For a simple, uniform planet - with parameters appropriate for the Earth:

See how the temperature profile evolves with time - the heat in the interior builds up. The gradient develops near the surface - which leads to stronger heat flux. The conductive cooling works its way lower and lower into the planet. But the radioactivity begins to slow down the heating.

In the following plots, a single variable is changed relative to the conditions of the first case.

You can see that we could build more complex models - multiple layers (core, mantle, crust) or have multiple radioactive isotopes with different half-lifes.

Two Computer Science majors, Ryan King and Jess Murphy, have made a Java applet of this model - Go to http://www.colorado.edu/engineering/ETH/projects/planetary_evo/Planet.htm - you need to have Java enabled on your browser. This is part of HW8.