Assume a Spherical FutureBaby…

Posted by Chad Orzel on May 20, 2008
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Welcome to today’s exciting episode of “How Big a Dork Am I?” Today, we’ll be discussing the making of unnecessary models:

i-6cc855b80ae87e3cb5ab039cdf6b8855-spherical_baby.jpg

In this graph, the blue points represent the average mass in grams of a fetus at a given week of gestation, while the red line is the mass predicted by a simple model treating the fetus as a sphere of uniform density with a linearly increasing radius.

The “model” was set up by taking the 40-week length reported at BabyCenter, and dividing by two to get an approximate radius for the spherical baby. Then I assumed that the actual radius increased linearly from zero to the final value, calculated the volume of the sphere, and multiplied by a constant density to get reasonable agreement between the model and the data.

If you take the numbers I put into this, and use them to estimate the mass of a cell in this model baby, you find that a cell with a volume of one cubic micron (10-18 m3) would have a mass of about 50 femtograms, which is kind of low, but remarkably good for such a silly model.

Oh, the things I will do to amuse myself…

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Comments

  1. #1 Aaron Bergman
    May 20, 2008

    That’s some serious procrastination you’ve got going there. Grading? Writing up a paper?

  2. #2 Chad Orzel
    May 20, 2008

    That’s some serious procrastination you’ve got going there. Grading? Writing up a paper?

    I was between a couple of meetings, and trying not to grade lab reports.

  3. #3 John Novak
    May 20, 2008

    NERD!

    (Also, there’s no error bars.)

  4. #4 dr. dave
    May 20, 2008

    Would this be a perfectly smooth, frictionless, spherical baby?

  5. #5 Captain Button
    May 20, 2008

    Inspired by the ultrasound pictures downblog:

    What is the mass of the Star Child at the end of 2001: A Space Odyssey?

  6. #6 Romeo Vitelli
    May 20, 2008

    When is this kid due again? You have way too much time on your hands.

  7. #7 Daryl McCullough
    May 20, 2008

    Here I am being an ambulance-chaser theorist. Why should the radius increase linearly with time? Here’s my theory: because the rate at which nutrients can enter the fetus is proportional to the surface area (maybe). According to this theory, we would predict: (where M = mass, A = surface area, and t = time)

    dM/dt = k A

    where k is an empirically determined constant. Assuming that M scales as the cube of some characteristic length, R, and A scales as the square of R, then we have

    k1 d/dt (R^3) = k k2 R^2

    where k1 and k2 are two other constants. This
    simplifies to

    d/dt R = k k2/(3 k1)

    We can absorb k1, k2 and the factor 3 into
    the unknown constant k to get

    d/dt R = k

    Do I win? Isn’t that even dorkier than your original post?

  8. #8 Anonymous
    May 20, 2008

    Should reach 9.2 metric tons in 10 years. Hope your floors are reinforced.

  9. #9 Ian Durham
    May 20, 2008

    Congrats, by the way. It’s blissfully enjoyable until they learn to talk. Anyway, I actually think that is pretty darn cool, by the way.

  10. #10 Ian Durham
    May 20, 2008

    Congrats, by the way. It’s blissfully enjoyable until they learn to talk. Anyway, I actually think that is pretty darn cool, by the way (the graph).

  11. #11 Chad Orzel
    May 20, 2008

    John Novak: Also, there’s no error bars.

    They just don’t show up on the graph…
    The data are specified to +/- 1 gram. I’m sure that makes sense. Really.

    dr. dave: Would this be a perfectly smooth, frictionless, spherical baby?

    But of course. Can’t you tell from the ultrasounds?

    (Actually, Kate would probably beg to differ regarding the “frictionless” part…)

    Daryl: Here I am being an ambulance-chaser theorist. Why should the radius increase linearly with time?

    It’s not, really. You can tell from the graph– the slope of the model is too high early on, and too low later. A somewhat higher power would probably be a better fit– taking out the early part (which is fairly close to exponential, almost doubling every week) and the last three points, weeks 16-40 fit pretty well to t^4.

    Linear growth is the easiest thing to simulate, though.

    Yes, I just cranked that into Excel and had it do a power-law fit. God, I’m a dork.

  12. #12 miller
    May 21, 2008

    I would have tried to model it with the logistical equation.

  13. #13 Luke
    May 23, 2008

    Since your baby is expanding linearly with time, Ω → 0. I’m sure you and your wife are glad that his/her self-gravity isn’t significant.