Nuts & Volts

SCALE HEIGHT
The scale height of an atmosphere
is an indication of how fast its
density decreases with altitude. Just
how fast the atmospheric density
(pressure also works here) changes
with altitude depends on the current
atmospheric density. This relationship
between rate of change and its
current value is also seen in things
like interest (how much money your
savings account will make this year
depends on how much money you
have in it at the beginning of the year)
and population (how many rabbits
you’ll raise this year depends on how
many rabbits you start the year with).
In the mathematics of these kinds of
changes, a specific number keeps
popping up, the number 2.71828 ....
This number is called e and it is related
to rates of change just like pi is
related to circles.
To make the mathematics simpler,
physicists look at how fast the atmospheric
density of a moon or planet
decreases by a factor of e. The altitude
at which this occurs is called the scale
height of that atmosphere. On Earth,
the scale height is 8.5 kilometers for
the average atmosphere at the Earth’s
surface. For Titan, this number is
between 40 and 60 km and for Mars it’s
11.1 km. The scale height is also the
height of an atmosphere if it had a
uniform density equal to its surface
pressure. So the top of Earth’s atmosphere
would be at 8.5 km high if its
density didn’t decrease with altitude.
The scale height of an atmosphere
is affected by the gravity of the
moon or planet and the mass of the
gas in the atmosphere. Decreasing the
gravity of a moon or planet increases
the scale height of its atmosphere as
does having a lower mass gas in
the atmosphere. Having a stronger
gravity and/or a denser gas in the
atmosphere means the atmosphere
is more compressed towards the
surface, which means the scale height
is lower. The planet Neptune and the
moon Titan have roughly the same
scale heights, because Neptune’s
atmosphere is largely hydrogen and
helium, and Titan’s gravity is so low.
Mass and gravity are not the only
things to affect scale height. Changes
in temperature also affect the scale
height of an atmosphere by causing
the gas in an atmosphere to expand
or contract. So in general, the scale
height of an atmosphere is given at
various altitudes and for an average
temperature, rather than for the entire
atmosphere.
94 March 2006
■ FIGURE 6
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■ FIGURE 7
Altitude (feet)
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that will creep into this volume calculation
of the balloon. The first is that
the equator of the balloon will rise
higher above the camera as the balloon
expands. This means the diameter
of the balloon will be slightly larger
than is indicated in the photographs.
The second is that balloon’s skin
tension will compress the balloon into
a smaller volume than air temperature
and pressure would like to make it.
To close out this Martian exercise, I
calculated what the balloon would see
at 48,000 feet above the Martian surface.
First, the horizon will be depressed
by 3.8 degrees. In other words, the horizon
will be 3.8 degrees lower than it is
from the surface of Mars. So from one
horizon, over the zenith, to the opposite
horizon, the sky will span 187.6 degrees,
Air Temperature (Titan)
70 75 80 85 90 95 100
Temperature (k)
Air Pressure (Titan)
200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600
Pressure (mb)
rather than the 180 degrees that it does
from the surface of Mars.
Second, from the balloon’s perspective,
the distance to the horizon will be
132 miles away from the point beneath
the balloon. On Earth, a balloon at
48,000 feet will see a horizon that is 268
miles away from the point beneath the
balloon. The greater distance on Earth is
due to Earth’s larger diameter and less
strongly curved surface.
BALLOONS OVER TITAN
I couldn’t find an equation for the
standard atmosphere for Titan. So in
place of an equation, I took measurements
from a chart of Titan temperature
and pressure and wrote them into
a spreadsheet. I then added a trend-