TABULA/LaunchElephant

From J Wiki
Jump to: navigation, search

How many pounds of ugly fat will launch an elephant into low-earth orbit?

This is how to go about solving the problem using TABULA.

Fat stores energy in your body. It's a very efficient store: a pound of fat yields a lot of energy. Which has implications if you want to try burning it off with exercise.

This leads to an intriguing question, which gives us the title of this article.

African elephants are beloved of journalists. If they have a large mass to write about, say approximately 100 tons, they think you'll appreciate it better if they describe it as "thirty African elephants". As if you had one these beasts standing on your bathroom scales at weekends.

Never mind. TABULA lets you express mass in African elephants if you so wish. There's even a special unit of mass defined for your convenience: [elephant], equal to some representative weight of an adult beast (3400 kg).

Method

Assume low-earth orbit just grazes the surface, so the total distance travelled in one orbital period is roughly the earth's circumference. If we divide that by the orbital period, that gives us the velocity of the elephant, from which we get the kinetic energy, which is the minimum energy needed to launch it.

Start with an empty t-table

To start a new t-table, click Tool00.jpg

Give it a name. Select the first (and only) line and enter: launch an elephant into orbit.

The orbital period

The TABULA formula library has a formula for orbital period. Let's find it.

Click tab: "Functs". Enter in the input field: orbit.

See the formula for orbital period (it looks like this):

PI2*sqrt(a/g) : g(grav),a(m)  [s]	earth orbital period

It comes into the t-table, bringing with it these feeder items:

  • the acceleration due to gravity: grav
  • the orbital radius: a.
launch an elephant into orbit
┌  1        1.000 grav  g:feeder
├  2        1.000 m     a:feeder
└> 3 @      2.006 s     earth orbital period

Feeder g comes with earth-gravity units [grav] already specified, plus the value: 1. This is strictly only valid at the earth's surface, but near enough to apply to low-earth orbit too. So we let that stand.

Let's enter the earth's radius into feeder a. We could enter it by hand, but instead let's use a value from the library of constants: UUC.

Click tab: "Consts". Enter in the input field: radius.

We see: equatorial radius of earth. Select it, and press "Append".

Now to merge it with feeder a we must move it above the said item.

Click the new item to select it (it should already be selected).

Now click Tool13.jpg to move the item up step-by-step until it becomes item 2.

Now select both items 2 and 3 (the old item 2) and click Tool25.jpg

You now see:

launch an elephant into orbit
┌  1        1.000 grav  g:feeder
├  2        1.000 eq.r  equatorial radius of earth!
└> 3 @   5065.587 s     earth orbital period

which has calculated for us the orbital period in seconds.

Let's convert it to minutes.

Click item (3) and select min from the Units dropdown.

We see: 84.426 min.

Note: Wikipedia declares "Low Earth orbit" to have a period of 89 - 128 min. It looks like we're on the right track.

Let's find what speed the orbiting body is travelling at. For that we need another formula: one for the circumference calculated from the radius.

Click tab: "Functs". Enter in the input field: circum.

Click the line: circumference of circle and click "Append".

The item comes in with a single feeder r.

We need to merge feeder r with item 2: equatorial radius of earth. As it happens, they are in the correct order to do so.

Items are merged downwards, meaning that the value and units of the uppermost item are the ones kept.

Select items 2 and 4 and click Tool25.jpg

You now see:

launch an elephant into orbit
  ┌  1        1.000 grav  g:feeder
┌ ├  2        1.000 eq.r  equatorial radius of earth!
│ └> 3       84.426 min   earth orbital period
└>   4 @    4.008E7 m     circumference of circle

Thus: 4.008E7 m is the distance travelled in one orbit, which for low-earth orbit is very little more than the Earth's circumference. If we divide this distance by earth orbital period, we get the speed of the orbiting object.

Select items 3 and 4, hold down Shift and click Tool08.jpg

Holding down Shift ensures that we get {4}/{3} and not {3}/{4}.

Speed-units of "metres per minute" are not very common. Let's convert to more familiar units, say: [km/h].

There are several methods of doing this.

Method 1: Select item 5: {4}/{3} and click Tool27.jpg

This converts the units to SI units, namely [m/s] here.

This still isn't quite what we want.

Method 2: Let's see what the Units dropdown gives us.

This will vary with the contents of the constants library, but it may only show us: [m/s] and [m/h] (as SI units).

Oh well. Select [m/h] in the Units dropdown.

Now look at the menu: Scale. This shows us the standard SI prefixes such as "kilo-", "mega-", "milli-"... in fact all of them, from "yocto" to "yotta".

Select menu: Scale > kilo- [*1000]

The units change to [km/h]

Method 3: This is the quickest way, provided you know how to write down the units you want.

Select the item to be changed, and in the input field simply enter the new units: km/h

However, if you mis-type the units, they may be interpreted as a new name for the item. If that happens, simply click Tool04.jpg and try again.

To avoid units being mistaken for a new name, enclose them in brackets: [km/h]

If you've mis-typed them, say as: [km/hour], you'll see an error message: >>> bad units: km/hour.

Give it a name. Enter: 'speed in orbit

You'll confuse yourself if you leave it as {4}/{3} since item numbers will change.

You now see this:

launch an elephant into orbit
    ┌  1        1.000 grav  g:feeder
  ┌ ├  2        1.000 eq.r  equatorial radius of earth!
┌ │ └> 3       84.426 min   earth orbital period
├ └>   4      4.008E7 m     circumference of circle
└>     5 @  28480.540 km/h  speed in orbit

The kinetic energy of a flying elephant

Now to launch our elephant.

Click tab: "Consts". Enter in the input field: elephant.

Click the line: standard African elephant and click "Append".

Click tab: "Functs". Enter in the input field: kinetic.

Click the line: kinetic energy and click "Append".

You now see:

launch an elephant into orbit
    ┌  1        1.000 grav      g:feeder
  ┌ ├  2        1.000 eq.r      equatorial radius of earth!
┌ │ └> 3       84.426 min       earth orbital period
├ └>   4      4.008E7 m         circumference of circle
└>     5    28480.540 km/h      speed in orbit
       6        1.000 elephant  standard African elephant!
    ┌  7        1.000 kg        m:feeder
    ├  8        1.000 m/s       v:feeder
    └> 9 @      0.500 J         kinetic energy

We need to merge the feeder items using Tool25.jpg ...

  • 6 and 7 (mass units)
  • 5 and 8 (speed units)

WARNING: the item numbers will change as you do so.

You now see:

launch an elephant into orbit
    ┌  1        1.000 grav      g:feeder
  ┌ ├  2        1.000 eq.r      equatorial radius of earth!
┌ │ └> 3       84.426 min       earth orbital period
├ └>   4      4.008E7 m         circumference of circle
└>  ┌  5    28480.540 km/h      speed in orbit
    ├  6        1.000 elephant  standard African elephant!
    └> 7 @   1.064E11 J         kinetic energy

How fat is the elephant?

First let us decide how much of our elephant consists of fat.

Let's start with 50% (which may represent a rather unhealthy elephant).

Click Tool11.jpg to create a dimensionless item.

Give it a name. Enter: 'percentage of fat in elephant

Now enter 50 %

Be sure to put a space before % or TABULA will make it the item name.

The last line now looks like this:

       8       50.000 %         percentage of fat in elephant

Multiply the percentage by item 6: standard African elephant by selecting items 6 and 8 and clicking Tool07.jpg

Then convert to SI-units by clicking Tool27.jpg

Give it a name. Enter: 'mass of fat in elephant

Scientists are careful not to say weight when they mean mass. (In orbit, an elephant is weightless.)

You now see:

launch an elephant into orbit
    ┌  1        1.000 grav      g:feeder
  ┌ ├  2        1.000 eq.r      equatorial radius of earth!
┌ │ └> 3       84.426 min       earth orbital period
├ └>   4      4.008E7 m         circumference of circle
└>  ┌  5    28480.540 km/h      speed in orbit
  ┌ ├  6        1.000 elephant  standard African elephant!
  │ └> 7     1.064E11 J         kinetic energy
  ├    8       50.000 %         percentage of fat in elephant
  └>   9 @   1700.000 kg        mass of fat in elephant

The energy in fat

Now let's discover the energy in a kilogram of fat.

Create a new item to represent 1 kg. Click Tool11.jpg and enter: 1 kg

Now what is the calorific value of a kilogram of fat? It doesn't matter from what animal or plant the fat comes, the calorific value is roughly the same. TABULA knows its value.

Click tab: "Consts". Enter in the input field: fat.

Click the line: energy content of fat and click "Append".

Now multiply the last 2 items. Select them and click Tool07.jpg

Change the units of {10}*{11} to [J] by using the Units dropdown, or by entering: J

Give it a name. Enter: 'energy in given mass of fat

Explanation: In spite of its name, item 11 is not in energy units. It can't be, because 2 kg of fat will have twice the energy content of 1 kg of fat. It needs multiplying by a mass quantity to turn it into energy units, eg [J].

You now see:

launch an elephant into orbit
    ┌   1        1.000 grav      g:feeder
  ┌ ├   2        1.000 eq.r      equatorial radius of earth!
┌ │ └>  3       84.426 min       earth orbital period
├ └>    4      4.008E7 m         circumference of circle
└>  ┌   5    28480.540 km/h      speed in orbit
  ┌ ├   6        1.000 elephant  standard African elephant!
  │ └>  7     1.064E11 J         kinetic energy
  ├     8       50.000 %         percentage of fat in elephant
  └>    9     1700.000 kg        mass of fat in elephant
    ┌  10        1.000 kg        unit
    ├  11        1.000 ener.fat  energy content of fat!
    └> 12 @    3.766E7 J         energy in given mass of fat

Now we want to link the feeder (item 10) with mass of fat in elephant (item 9).

Select items 9 and 10. Click Tool25.jpg

You now see:

launch an elephant into orbit
    ┌   1        1.000 grav      g:feeder
  ┌ ├   2        1.000 eq.r      equatorial radius of earth!
┌ │ └>  3       84.426 min       earth orbital period
├ └>    4      4.008E7 m         circumference of circle
└>  ┌   5    28480.540 km/h      speed in orbit
  ┌ ├   6        1.000 elephant  standard African elephant!
  │ └>  7     1.064E11 J         kinetic energy
  ├     8       50.000 %         percentage of fat in elephant
  └>┌   9     1700.000 kg        mass of fat in elephant
    ├  10        1.000 ener.fat  energy content of fat!
    └> 11 @   6.402E10 J         energy in given mass of fat

Getting two items to take roughly the same value

We now have two energy items to compare:

  • Item 7: the kinetic energy of the orbiting elephant
  • Item 11: energy in given mass of fat -- viz. the fat in the elephant.

Let's call the kinetic energy KE, and the elephant's fat-energy FE.

Given an efficient rocket engine, we could use the fat in the elephant to try to launch it into orbit. Will there be enough?

Let's divide FE by KE to see how close we are.

Select the two energy items. Shift-click Tool08.jpg

The result is a new item: 0.602 J/J

Give it a name. Enter: 'energy ratio

Now [J/J] is the same as [/], which is dimensionless. So we should be able to express energy ratio as a percentage.

Click the Units dropdown. We do indeed see: %. Select it.

The result is: 60.165 % -- not quite enough.

But hey! -- what if we increase the percentage of fat in the elephant?

Select the item: percentage of fat in elephant and click repeatedly on Tool22.jpg to step its value by 1 each time.

By the time we get to 83%, we see that energy ratio has reached over 99%.

Forcing two items to take exactly the same value

Let's not be satisfied with that. Let's make the two energy items exactly the same, and see what percentage of fat the elephant needs to contain.

For this to work, we only want item 8 to change: percentage of fat in elephant. So we must hold all the items we don't want to change.

We only need to hold non-calculated items: the calculated items will then be held too.

The non-calculated items are all those with no arrowheads pointing to them. They are: 1, 2, 6, 8, 10. Notice that 2, 6, 10 have ! at the end of their names. This means they are held already.

This leaves only 1 and 8 for us to decide to hold or not.

You can toggle the hold-status of any item by selecting it and clicking Tool18.jpg
or by menu: Command > Toggle Hold

If there is a final ! then it disappears. If there is no final ! then one appears.

Hold item 1. Select it and click Tool18.jpg

Leave item 8 to change if it can.

Select item: energy ratio and enter: 100

signifying: 100%

You now see:

launch an elephant into orbit
    ┌   1        1.000 grav      g:feeder!
  ┌ ├   2        1.000 eq.r      equatorial radius of earth!
┌ │ └>  3       84.426 min       earth orbital period
├ └>    4      4.008E7 m         circumference of circle
└>  ┌   5    28480.540 km/h      speed in orbit
  ┌ ├   6        1.000 elephant  standard African elephant!
┌ │ └>  7     1.064E11 J         kinetic energy
│ ├     8 @     83.105 %         percentage of fat in elephant
│ └>┌   9 @   2825.571 kg        mass of fat in elephant
│   ├  10        1.000 ener.fat  energy content of fat!
├   └> 11 @   1.064E11 J         energy in given mass of fat
└>     12 @    100.000 %         energy ratio

The @-flag shows all the items that have changed as a result.

As a result, we see that Item 8 has changed to 83.105 %

This is close to the 83% we saw before. But this time it is precise.

A final question

We finish with a final question:

Does this proportion of fat to body-weight change for different weights of elephant?

Well, let's try it and see.

Change item 6: standard African elephant from 1.0 to 1.1 (a 10%-heavier elephant).

There's no need to change the hold-status. Hold only applies to back-fitting.

Notice that item 8: percentage of fat in elephant remains steadfastly firm at 83.105%.

Change item 6: standard African elephant from 1.0 to 2.0

Item 8 still doesn't change.

This stands to reason. It doesn't matter whether we fire off an elephant that's twice as heavy, or two elephants glued together, or two elephants, one after the other. The energy requirement doesn't change whether the orbiting object(s) are glued or not.

Otherwise NASA would get through a helluva lot of glue!

It doesn't have to be an elephant at all. We can substitute (the weight of) a heavy man: 130 kg.

Select item 6 and convert to SI-units by clicking Tool27.jpg

We see the elephant's weight: 3400 kg.

Enter: 130 kg (the representative mass of an obese man).

No need to relabel the item. The man can be an honorary elephant for now.

Item 8 still doesn't change: 83.105%.

A moment's reflection tells us that's what we ought to expect. The kinetic energy of a payload in low-earth orbit is proportional to its mass, surely? For if it breaks up (without actually flying apart), the pieces will keep together (more or less) -- or at least stay in the same orbit.

Otherwise those spacewalks we see on TV would be impossible to do!

It also leads us to some intriguing corollaries:

1. Elephants may look fat, but they're unlikely to be 83% fat. But for an obese person, 83% of his body mass might well consist of fat. Enough fat to launch him into low-earth orbit if it were burnt as rocket fuel in an efficient engine.

2. Moreover this simple calculation doesn't take into account the fact that as the fat gets burnt, his body mass will diminish, resulting in an even higher orbit.

3. If an obese person aims to run off a sizeable proportion of his fat, he'll have to run very fast!

Ah well... back to the diet sheet.


Contributed by Ian Clark