Module 6: Atmospheres & Fluids

DENSITY OF SOLID WORLDS


Adapt the "Density of Icy Moons" (from Mod 2) activity for rocky worlds, to determine the density of of Metal-Core Worlds.

For this activity, we would like your group to design and carry out an inquiry based activity that is an adaptation of the "Icy Worlds" activity to address students who may have this question (and many of you mentioned this back in Module 2 when we did the "Icy Worlds" activity). Please keep in mind that this is a course for teachers, and there are two goals of this course. One is to learn new content, and the other is to be able to present the content to your students in the classroom in the form of creative lesson plans.

As indicated in the Image Processing activity, we have tried to move from "Instructor-led" activities during the beginning of the course, to more "inquiry based" activities at this point. This strategy is based on National Science Standards for Science Teaching.

For this activity, design and carry out an inquiry activity designed to calculate the fraction of rock and metal for the terrestrial worlds and at least one other rocky body in our Solar System. Compare your results to Figure 9.5, p. 230, and to common sense. (It is useful to calculate the fractional core radius for rough comparison to the figure.)

There are some ambiguities - rock can have quite a range of densities, and at least one world gives strange results if you use the "typical" value. There's more than one answer - sometimes you just don't know things, and the best you can do is estimate or determine a range of possibilities. Learning to deal with imperfect and incomplete knowledge - and still get results - is also part of science, and especially something you have to deal with when addressing real inquiries.

Turn in the values you used, the results, and your group's most significant comments.

Following are some notes which are included to help you carry out this investigation:

Method: assume the world is only made of two things (metal & rock -or- rock & ice). Then:

  • D_world = D1*F1 + D2*F2
  • Where D1= density of material 1, F1 = volume fraction of world made of material 1
  • F1 + F2 = 1 (or 100%)

Algebra:

  • D_world = D1*F1 + D2*(1-F1)
  • D_world = D1*F1 + D2*1 -D2*F1
  • D_world = D2*1 + (D1-D2)*F1
  • D_world - D2 = (D1 - D2)*F1
  • (D_world - D2) / (D1 - D2) = F1

Result:

  • F1 = (D_world - D2) / (D1 - D2) (Multiply by 100 to get percentage, if you prefer)
  • F2 = 1 - F1

A spreadsheet is remarkably handy for these calculations, and makes it easier to play with the input density values. I strongly recommend using a spreadsheet if you're not going to solve the problem graphically. In a spreadsheet, it also becomes easy to calculate fractions of a three-component world, using a range of values for one of the components, if you wish.

Alternative: plug in 0.0, 0.1, 0.2, ... , 0.9, 1.0 for F1 in the first equation, and 1-F1 for F2. Use the densities of rock and metal. Graph the results as in the ice & rock activity, and then read the fraction of metal from the graph.

See below for useful density values. The fractional radius of the core is F1^(1/3) (cube root of F1, if you've used the denser material for F1 and left the result as a fraction).

Densities:

From the original activity:

  • Rock: 3.5 g/cm^3; Ice: 0.9 g/cm^3
  • But the teacher instructions also note that: water-bearing minerals: 2.5-3.0 g/cm^3; compressed rock > 3.5 g/cm^3; ammonia & other ices are of varied densities.

From Cosmic Perspective, p. 289:

  • Hydrogen (gas) at 1 bar and 125K : 0.0002 g/cm^3
  • (transition to liquid) at 5E5 bars and 2000K: 0.5 g/cm^3
  • (transition to metallic) at 2E6 bars and 5000K: 1.0 g/cm^3
  • Jovian core of rock, metal, hydrogen compounds at 1E8 bar and 20,000K: 25 g/cm^3
  • See also p. 293.

From Zeilik, Astronomy, the Evolving Universe:

  • Water (liquid): 1.0g/cm^3
  • rocks: 2 - 4 g/cm^3
  • pure iron: 7.8 g/cm^3
  • Rocks near Earth's surface, average = 2.4 g/cm^3
  • Estimates of density at Earth's core: 12.0 g/cm^3


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last updated 5/4/02