Space Math: Estimating the Temperatures of Exoplanets
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Space Math: Estimating the Temperatures of Exoplanets

Topic: Math

Body:

Mission:

Science Education Standards: Physical Science -- Content Standard B: Motions and Forces

Short Description: Students review the basic properties of ellipses by exploring the orbits of newly-discovered planets orbiting other stars. They also use a simple formula to determine the temperatures of the planets from their orbits.

Source: Space Math (GSFC)

Because many exoplanets orbit their stars in elliptical paths, they experience large swings in temperature. Generally, organisms can not survive if water is frozen (0 C = 273 K) or near its boiling point (100 C or 373 K). Due to orbital conditions, this very narrow 'zone of life' may not be possible for many of the worlds detected so far.

Problem 1 - Complete the table below by calculating

A) The semi-minor axis distance B = A(1-e2)

B) The perihelion distance Dp = A(1-e)

C) The aphelion distance, Da = B(1+e)

Problem 2 - Write the equation for the orbit of 61 Vir-d in Standard Form.

Problem 3 - The temperature of a planet similar to Jupiter can be approximated by the formula below, where T is the temperature in Kelvin degrees, and R is the distance to its star in AU. Complete the table entries for the estimated temperature of each planet at the farthest 'aphelion' distance Ta, and the closest 'perihelion;' distance Tp.

T(R) = 250 / √R

Problem 4 - Which planets would offer the most hospitable, or the most hazardous, conditions for life to exist, and what would be the conditions be like during a complete 'year' for each world?

 Planet A (AU) B (AU) e Period (days) Dp (AU) Da (AU) Ta (K) Tp (K) 47 UMa-c 3.39 0.22 2190 61 Vir-d 0.47 0.35 123 HD106252-b 2.61 0.54 1500 HD100777-b 1.03 0.36 383 HAT-P13c 1.2 0.70 428

Problem 1 - See table below:

 Planet A (AU) B (AU) e Period (days) Dp (AU) Da (AU) Ta (K) Tp (K) 47 UMa-c 3.39 3.3 0.22 2190 2.6 4.1 61 Vir-d 0.47 0.4 0.35 123 0.3 0.6 HD106252-b 2.61 2.2 0.54 1500 1.2 4.0 HD100777-b 1.03 1.0 0.36 383 0.7 1.4 HAT-P13c 1.2 0.9 0.70 428 0.4 2.0

Problem 2 - Write the equation for the orbit of 61 Vir-d in Standard Form.

Answer: A = 0.47 and B = 0.4

So 1 = x2 / 0.47 + y2 / 0.40 and also 188 = 40 x2 + 47 y2

Problem 3 - See table below:

 Planet A (AU) B (AU) e Period (days) Dp (AU) Da (AU) Ta (K) Tp (K) 47 UMa-c 3.39 3.3 0.22 2190 2.6 4.1 154 123 61 Vir-d 0.47 0.4 0.35 123 0.3 0.6 452 314 HD106252-b 2.61 2.2 0.54 1500 1.2 4.0 228 125 HD100777-b 1.03 1.0 0.36 383 0.7 1.4 308 211 HAT-P13c 1.2 0.9 0.70 428 0.4 2.0 417 175

Problem 4 - Which planets would offer the most hospitable, or most hazardous, conditions for life to exist, and what would be the conditions be like during a complete 'year' for each world?

Answer: For the habitable 'water' range between 273K and 373K, none of these planets satisfy this minimum and maximum condition. They are either too hot at perihelion 'summer' or too cold at 'winter' aphelion.

Only HD100777-b during perihelion is in this temperature range during 'summer', at a temperature of 308 K (35 C). During 'winter' at aphelion, it is at -62 C which is below the freezing point of water, and similar to the most extreme temps in Antarctica.

Note: These temperature calculations are only approximate and may be considerably different with greenhouse heating by the planetary atmosphere included.