Space Math: Hubble: The Changing Atmosphere of Pluto
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Space Math: Hubble: The Changing Atmosphere of Pluto

Topic:

Mission:

Science Education Standards:

Earth and Space Science

Energy in the Earth System

• Heating of Earth's surface and atmosphere by the sun drives convection within the atmosphere and oceans, producing winds and ocean currents.
• Global climate is determined by energy transfer from the sun at and near the Earth's surface. This energy transfer is influenced by dynamic processes such as cloud cover and the Earth's rotation, and static conditions such as the position of mountain ranges and oceans.

The Origin and Evolution of the Earth System

• Interactions among the solid Earth, the oceans, the atmosphere, and organisms have resulted in the ongoing evolution of the Earth system.

Short Description: Based on a recent press release, students determine the aphelion and perihelion of Pluto's elliptical orbit using the properties of ellipses, then calculate the temperature of Pluto at these distances to estimate the thickness of Pluto's atmosphere and its changes during its orbit around the sun.

Source: Space Math (GSFC)

Recent Hubble Space Telescope studies of Pluto have confirmed that its atmosphere is undergoing considerable change, despite its frigid temperatures. Let's see how this is possible!

Problem 1 - The equation for the orbit of Pluto can be approximated by the formula 2433600=1521x2+1600y2. Determine from this equation, expressed in Standard Form, A) the semi-major axis, a; B) the semi-minor axis, b; C) the ellipticity of the orbit, e; D) the longest distance from a focus called the aphelion; E) the shortest distance from a focus, called the perihelion. (Note: All units will be in terms of Astronomical Units. 1 AU = distance from the Earth to the Sun = 1.5x1011 meters).

Problem 2 - The temperature of the methane atmosphere of Pluto is given by the formula

T (R) = [ L(1 - A) / 16 π σ R2 ]1/4 degrees Kelvin (K)

where L is the luminosity of the sun (L=4 x 1026 watts); σ is a constant with a value of 5.67x10-8, R is the distance from the sun to Pluto in meters; and A is the albedo of Pluto. The albedo of Pluto, the ability of its surface to reflect light, is about A = 0.6. From this information, what is the predicted temperature of Pluto at A) perihelion? B) aphelion?

Problem 3 - If the thickness, H, of the atmosphere in kilometers is given by H(T) = 1.2 T with T being the average temperature in degrees K, can you describe what happens to the atmosphere of Pluto between aphelion and perihelion?

In Standard Form 2433600=1521x2+1600y2 becomes 1 = x2 / 1600 + y2 / 1521 = x2 / a2 + y2 / b2

Then A) a = 40 AU and B) b=39 AU. C) The ellipticity e = (a2- b2)1/2/a = 0.22. D) The longest distance from a focus is just a(1 + e) = 40(1+0.22) = 49 AU. E) The shortest distance is just a(1-e) = (1-0.22)(40) = 31 AU. Written out in meters we have a= 6x1012 meters; b= 5.8x1012 meters; aphelion = 7.35x1012 meters and perihelion = 4.6x1012 meters.

Problem 2 - Answer: For R in terms of AU, the formula simplifies to

T (R) = [ 4x1026(1 -0.6) / 16(3.14)(5.67x10-8)(1.5x1011)2R2 ]1/4 so T (R) = 223 / √R degrees K

A) For a perihelion distance of 31 AU we have T = 223/(31)1/2 = 40 K; B) At an aphelion distance of 49 AU we have T = 223/(49)1/2 = 32 K. Note: The actual temperatures are about higher than this and average about 50K.

Problem 3 - Answer: At aphelion, the height of the atmosphere is about H=1.2x(32) = 38 kilometers, and at perihelion it is about H=1.2x(40) = 48 kilometers, so as Pluto orbits the sun its atmosphere increases and decreases in thickness.

Note: In fact, because the freezing point of methane is 91K, at aphelion most of the atmosphere freezes onto the surface of the dwarf planet, and at aphelion it returns to a mostly gaseous state. This indicates that the simple physical model used to derive H(T) was incomplete and did not account for the freezing-out of an atmospheric constituent.