Atmosphere the Aeronauts Follows the Adventures of James Glaisher, a Scientist, and Amelia Wren, a Flamboyant Aeronaut Who Lost Her Husband in a Balloon Accident

Copyrights Prof Marko Popovic 2021 Atmosphere The Aeronauts follows the adventures of James Glaisher, a scientist, and Amelia Wren, a flamboyant aeronaut who lost her husband in a balloon accident. The pair, fighting against thunderstorms, wind, hailstones and rain as they ascend higher and higher, achieve something phenomenal: they travel to heights no man or woman has ever reached before. In The Aeronauts, meteorologist James Glaisher (Redmayne) presents his theories of how a gas balloon expedition could be key to predicting the weather—a science still in its infancy in the 1860s—and asks for funding for the expedition. His peers respond emphatically: “We are scientists, not fortune tellers.” But Glaisher doesn’t give in. In the movie, the pair breaks the world record for altitude after reaching a height of 36,000 feet. https://www.imdb.com/title/tt6141246/?ref_=vp_vi_tt Glaisher did in fact exist, and he did break the record for traveling higher than any person, but he did so with fellow scientist Henry Tracy Coxwell rather than the fictional character of Amelia Wren. On Sept. 5, 1862, the two men, equipped with pigeons (as in the film), a compass and thermometers, took to the skies and broke the world record for the highest any human had been in a balloon. Gleisher passed out around at 8,800 meters (28,900 feet) before a reading could be taken. Estimates suggest that he rose to more than 9,500 metres (31,200 feet) and as much as 10,900 metres (35,800 feet) above sea level. Coxwell lost all sensation in his hands. The valve-line had become entangled so he was unable to release the mechanism; with great effort, he climbed onto the rigging and was finally able to release the vent before losing consciousness. This allowed the balloon to descend to a lower altitude. In real life, Glaisher was indeed an influential scientist—he made 28 ascents between 1862 and 1866, recording observations that were crucial to our understanding of weather. Among his discoveries were the fact that wind changes speed at different altitudes, and the way raindrops form and gather moisture. Science has, of course, advanced significantly since Glaisher’s time. The kinds of scientific measurements he performed using thermometers, barometers and hygrometers are now made in unmanned meteorological balloons. Modern balloons generally contain electronic equipment such as radio transmitters, cameras, or satellite navigation systems, such as GPS receivers. In 2002, a balloon named BU60-1 reached a record altitude of 53.0 km (32.9 mi; 173,900 ft). How Planes Measure Altitude? How Planes Measure Altitude? Altimeter displays indicated altitude and this is what is currently used for all traffic separation in US and Canada. Indicated altitude is pressure altitude corrected for local atmospheric conditions. The correction is done by entering the altimeter setting given by Air Traffic Control or on an Automated Weather Observing System (AWOS). All aircrafts in a given area should be on the same altimeter setting so relative (altitude) separation is maintained. A GPS, on the other hand, measures absolute altitude off several ≥ 4 satellites. While potentially more accurate than pressure altitude, it does not provide the same relative separation from other aircrafts (since all aircrafts are using indicated altitude) PRESSURE ALTITUDE (typically from an aneroid barometer) (as a check, GPS and/or radio altimeter are also used) while monitoring temperature Atmospheric Constituents Dry air uniform composition in Homosphere (0 ~ 90 푘푚) 푚 푁푁22 × 14 푚푝 + 푁푂22 × 16 푚푝 + 푁퐴푟40 푚푝 = ≅ 푉 푉 푁 푚푝 proton mass = 0.781 × 28 + 0.210 × 32 + 0.009 × 40 푚 푉 푝 푛푚 number of moles 푁 푛 푁 푛 푀 푁퐴 Avogadro’s number = 28.948 푚 = 푚 퐴 28.948 푚 ≅ 푚 푉 푝 푉 푝 푉 푀 molar mass Recall the ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. 푝푉 = 푁푘푇 = 푛푚푅푇 푅 = 8,314.472 퐽 푘푚표푙−1 퐾−1 universal gas constant −23 2 −2 −1 푘 also 푘퐵 = Boltzmann constant = 1.38064852 × 10 푚 푘푔 푠 퐾 푁푘 = 푛푚푅 hence 푅 = 푁퐴푘 − 27 푚푝 proton mass 1.67262 × 10 푘푔 푛푚 number of moles 23 −1 푁퐴 Avogadro’s number 6.0221409 × 10 푚표푙 푀 molar mass (air) 2.91 × 10−2 푘푔 푚표푙−1 Variations with altitude 퐴 흆 풉 퐀 ∆풉 품 = 풑 풉 − ∆풉/ퟐ 푨 − 풑 풉 + ∆풉/ퟐ 푨 density , pressure , temperature ℎ + ∆ℎ , 푝 ℎ + ∆ℎ , 푇 ℎ + ∆ℎ 푑푝 g = − ℎ , 푝 ℎ , 푇 ℎ 푑ℎ ℎ − ∆ℎ , 푝 ℎ − ∆ℎ , 푇 ℎ − ∆ℎ 푁 푚 푁푁2 2 × 14 푚푝 + 푁푂2 2 × 16 푚푝 + 푁퐴푟 40 푚푝 = ≅ ≅ 푉 푉 푉 28.948 푚푝 푁 = 0.781 × 28 + 0.210 × 32 + 0.009 × 40 푚 푉 푝 푑푝 푁 푑ℎ 푑 ln 푝 28.948 푚푝푔 푀푔 푝 = 푘푇 ≅ 푘푇 = − 푘푇 = − = − 푉 28.948 푚푝 28.948 푚푝푔 1 푑ℎ 푘푇 푅푇 푑푝 = d ln 푝 푝 Variations with altitude – constant T approximation ℎ 1 1 1 1 푀푔 푝1 푀푔 න 푑 ln 푝 = − න 푑ℎ ln푝1 − ln푝0 = ln = − න 푑ℎ 0 0 푅푇 푝0 0 푅푇 1 푀푔 ׬ 푑ℎ − 0 푝1 = 푝0푒 0 푅푇 In the zeroth order (constant) approximation assume 푇 ℎ = 푇0 = 푐표푛푠푡 . 푀 푀푔 − ℎ −ℎ − ℎ1−ℎ0 푅푇 1 0 푝 − 푝 = 푝 [푒 푅푇0 − 1] Then 푝1 = 푝0푒 0 and 1 0 0 ∆ℎ 푅푇0 Scale − 퐻 where 퐻 = ∆푝 = 푝0 푒 − 1 푀푔 Height Variations with altitude – constant T approximation If one sets ℎ0 = 0, 푝0 = 푝 0 and renames 푝1 → 푝 as well as ℎ1 → ℎ then ℎ 푅푇0 − with scale factor 퐻 = 푝 ℎ = 푝0푒 퐻 푀푔 1 푑푝 푝 푝0 ℎ ℎ From here it also follows that = − = and = 푒− 퐻 = 푒− 퐻 푔 푑ℎ 푔퐻 푔퐻 0 However, temperature is not constant with altitude; scale factor varies with altitude For typical temperature distribution 퐻 0 ≅ 8.5 푘푚 퐻 5 푘푚 ≅ 7.5 푘푚 퐻 10 푘푚 ≅ 6.5 푘푚 Apparently barometric altimeters in planes use relationship that we just obtained to relate pressure and altitude. ℎ ℎ 푝0 푅푇0 푝0 푝0 푝 ℎ = 푝 푒− 퐻 → = ln → ℎ = ln = 푐푇 ln 0 퐻 푝 푀푔 푝 푝 Pilot uses airport’s altitude and current pressure (and temperature?) to calibrate altimeter which essentially translates to setting correct local value of scale factor. Points of concern: this formula is the result of simplest (wrong) model assuming constant temperature for the entire air column (hence changes in temperature alone may easily introduce 20-30% errors), value of 푔 is changing (<1%) with latitude, molar mass is changing depending on water vapor concentration (up to ~3% in tropics), also, air is entering barometric altimeter from an external port – hence the exact location of the port, plane’s geometry, and plane speed may introduce additional variability in estimating the plane’s altitude. Mess! 푣Ԧ 퐴 A 푣Ԧ 퐵 Pressure 1 B Pressure 2 Pressure 3 How accurate is the pressure reading? Important for air traffic separation. Any dependence on sensor placement (convention) and aircraft speed? For example, consider Bernoulli’s equation ? for adiabatic flow at less than Mach 0.3 풗 ? ? 푣2 푝 + + 푔ℎ = 푐표푛푠푡 ? ? 2 Variations with altitude – linear approximation In the first order (linear) approximation assume 푇 ℎ = 푇0 1 + 푎 ℎ − ℎ0 . ℎ 1 1 1 1 푑ℎ 1 푑ℎ 1 푑ℎ 1 푑 푏 + 푐ℎ 푏 + 푐ℎ 푐 න = න = න = න 푐 = ln 1 0 푇 0 푇0 1 + 푎 ℎ − ℎ0 0 푏 + 푐ℎ 0 푏 + 푐ℎ 푏 + 푐ℎ0 0 1 1 푑ℎ that is ׬ = ln 1 + 푎 ℎ − ℎ 푇0푎 . Hence 0 푇 1 0 푀 1 푀푔 푅푇 푎 ׬ 푑ℎ − ln 1+푎 ℎ1−ℎ0 0 − 푝1 = 푝0푒 0 푅푇 = 푝0푒 푀푔 − 푅푇 푎 푝1 = 푝0 1 + 푎 ℎ1 − ℎ0 0 Variations with altitude – linear approximation Again, if one sets ℎ0 = 0, 푝0 = 푝 0 and renames 푝1 → 푝 as well as ℎ1 → ℎ then 1 − 푀푔 1 푇 퐻푎 − 푅푇 푎 − 푝 = 푝0 1 + 푎ℎ 0 = 푝0 1 + 푎ℎ 퐻푎 = 푝0 푇0 It is easy to verify that this goes to our zeroth order result when 푎 → 0 1 1 푑푝 푝0 Similarly as before = − = 1 + 푎ℎ − 퐻푎−1 푔 푑ℎ 푔퐻 1 − −1 푝 1 푇 퐻푎 − 퐻푎−1 푝0 = = 0 1 + 푎ℎ = 0 with 0 = 1 + 푎ℎ 푔퐻 푇0 푔퐻 Let’s plot results for linear assumption in troposphere 150 퐾 200 퐾 250 퐾 300 퐾 Mean value for tropopause Lapse rate (9 푘푚 @ poles and 푑푇 − = −푎푇 = ퟕ. ퟕ 푲 풌풎−ퟏ 17 푘푚 @ equator; 푑ℎ 0 Temp.= −ퟖퟎ풐푪) assumed −ퟏퟏퟐ풐푭 assumed Temperature 5 Density 푝0 = 1.01235 × 10 푃푎 3 Pressure 0 = 1.225 푘푔/푚 푇0 = 293 퐾 20표퐶 68표퐹 note sensitivity on boundary condition (temperature @ tropopause) 150 퐾 200 퐾 250 퐾 300 퐾 Mean value for tropopause Lapse rate (9 푘푚 poles and 푑푇 −ퟏ − = −푎푇0 = ퟔ. ퟐ 푲 풌풎 17 푘푚 equator; 푑ℎ Temp.= −ퟔퟎ풐푪) assumed −ퟕퟔ풐푭 assumed Temperature 5 Density 푝0 = 1.01235 × 10 푃푎 3 Pressure 0 = 1.225 푘푔/푚 푇0 = 293 퐾 20표퐶 68표퐹 푑푇 휌 ℎ 푝 ℎ For smaller temperature change, = 푎 → 0, pressure and density vs altitude are more similar, → 푑ℎ 휌0 푝0 150 퐾 200 퐾 250 퐾 300 퐾 Mean value for tropopause Lapse rate 푑푇 (9 푘푚 poles and −ퟏ − = −푎푇0 = ퟒ. ퟔ 푲 풌풎 17 푘푚 equator; 푑ℎ Temp.= −ퟒퟎ풐푪) assumed −ퟒퟎ풐푭 assumed Temperature 5 Density 푝0 = 1.01235 × 10 푃푎 3 Pressure 0 = 1.225 푘푔/푚 푇0 = 293 퐾 20표퐶 68표퐹 How much estimated height varies if lapse rate is changing, i.e.

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