Planetary Temperature

The Planetary Temperature calculator estimates the temperature of a planet, either in our own solar system or an exoplanet (a planet circling another star); it's meant to help people interested in writing science fiction to design more realistic planets, or for anyone who's interested in exoplanet searches and the search for extraterrestrial life.  The algorithm includes a simple model to include the effects of Greenhouse gases in the atmosphere. 

INSTRUCTIONS: Choose units and enter the following:

• (L) Luminosity (brightness) of Star, in units where the sun's luminosity is 1
• (d) Average Distance of Planet from Star, in units where the Earth's average distance is 1 (i.e., in AU, or astronomical units)
• (a) Average Planetary Albedo (the fraction of light that the planet reflects in the visible region of the spectrum)
• (τ) Infrared Optical Depth of Planetary Atmosphere (a measure of how much heat the planet traps due to the Greenhouse effect)

Planetary Temperature (T):  The calculator returns the following:

• Relative light flux (compared to Earth) as a real number
• Temperature (no Greenhouse effect) in degrees Kelvin
• Temperature (including Greenhouse effect) in degrees Kelvin

The Math / Science

In the formula shown, σ is the Stefan-Boltzman constant (5.67x10^-8 W/m²K⁴). The relative flux,  the amount of light compared to the amount received at Earth's surface, is L/d²; the flux, F, is the relative flux times the solar constant (1361 W/m²).

This model is only approximate, particularly in how it handles the Greenhouse effect.  It also won't work for planets that are so close to the star that they are tidally locked (that is, with one side always facing the star, and one side always facing away from it.). 

Infrared Optical Depth of Planetary Atmosphere (τ)

The infrared optical depth of a planetary atmosphere refers to the measure of how effectively the atmosphere absorbs and scatters infrared radiation at different wavelengths. In simpler terms, it's a way to quantify how opaque the atmosphere is to infrared light.  When sunlight reaches a planet, it warms the surface, and the planet then emits infrared radiation as heat. This emitted infrared radiation interacts with the gases and particles in the atmosphere. Some of this radiation is absorbed by the atmosphere, while some is scattered or reflected back into space.  The optical depth is a measure of how much of this infrared radiation is absorbed or scattered by the atmosphere before it reaches space. A higher optical depth indicates that more radiation is absorbed or scattered, meaning the atmosphere is more effective at trapping heat. This is the basis for the greenhouse effect on Earth and other planets with significant atmospheres.

Scientists use measurements of infrared radiation at different wavelengths to determine the optical depth of a planetary atmosphere. This information is crucial for understanding the energy balance of a planet and its climate system.

Higher values of τ = more Greenhouse warming.  Earth, which has a moderate GH effect, has τ of about 0.8; Venus, which has a runaway Greenhouse effect, has τ of about 100.  Again, take these calculations with a big grain of salt - you need to model atmospheric absorption in a much more detailed way to be able to accurately predict real planetary temperatures.

Exoplanet Functions:

• Separation from Mass and Period
• Speed of Circular Orbit
• Mass of Exoplanet from Mass and Speed of Star and Planet Speed
• Mass of Exoplanet from Stellar Mass and Radius around barycenter and planetary orbit radius
• Mass from Period and Separation
• Mass from Speed and Separation
• Radius from Speed and Period
• Speed from Delta Lambda and Lambda
• Distance from Apparent and Absolute Magnitude
• Flux Ratio from Magnitudes
• Planetary Temperature
• Goldilocks Zone

Weather (Meteorology) Calculators

• Heat Index: Approximates the heat index based on the temperature and relative humidity
• Canadian Humidity Index (HUMIDEX): Computes the Canadian Humidity Index (HUMIDEX) based on the dry-bulb ambient temperature and the dew-point temperature.
• Australian Apparent Temperature (AT): Computes the apparent temperature based on the dry ambient temperature, the relative humidity and the wind speed at ten meters.
• Relative Humidity: Computes the relative humidity based on the actual density of vapor and the saturated density.
• Dew Point from Relative Humidity: Computes the dew point based on the relative humidity and ambient temperature.
• Summer Simmer Index: Computes the Summer Simmer Index based on the air temperature and relative humidity.
• Wind Chill Index (North America): Computes the wind chill index based on the air temperature and the velocity of the wind.
• Wet-Bulb Globe Temperature (WBGT): Computes the wet-bulb globe temperature index based on dry-bulb (air) temperature, globe thermometer temperature and the wet-bulb temperature.
• Humature Index: Estimates the humidity index based on the temperature and dew point.
• Antoine Equation: Computes the apparent vapor pressure of pure substances based on temperature and coefficients for the substance.
• Barometric Formula (Tropospheric) calculator computes the normal barometric pressure based on the altitude (h) using the Exponential Atmosphere formula.
• Planetary Temperature: Estimates a planet's temperature based on the luminosity of its star, distance to the star, average albedo and infrared optical depth of atmosphere.
• Snow Water Equivalent (SWE): Computes the volume of liquid water contained in rain or snow pack defined by area, depth and snow type or rain.

Astronomical Units

Because of the enormity of space and the size of the objects studied, the field of astronomy employs units not commonly used in everyday life.  Nonetheless, these units do translate into common units at a grand scale, and vCalc provides automatic conversions between units for calculator inputs and answers via the pull-down menus.  The following is a brief description on the distance, mass and time units employed in the field of astronomy

Astronomy Distance Units

Astronomical Unit (au): Within our solar system, a common measure of distance is au, which stands for astronomical units.  A single astronomical unit is the mean distance from the Sun's center to the center of the Earth.

Light Travel in Time: Light is a primary observable when studying celestial bodies.  For this reason, the distance to these objects are measured in the amount of time it would take light to travel from there to the Earth.  We can say that an object is one light-year away, and that means that the object is at a distance where it took an entire year for light from the object to travel to Earth.  Since the speed of light is 299,792,458.0 meters per second, one can compute the distance equal to a light year as follows:

1 light year = 299,792,458.0 (meters / second) x 31,536,000 (seconds / year) = 9,460,528,405,000,000 meters

The same exercise can be used for light traveling shorter periods of times, light seconds, light minutes, light hours and light days.  Since even these units are not enough when computing distances across the universe, there is also a light relative distance of kilo-light years (1000 light years), or the distance light travels in a thousand years!

Angle Shift Seen from Earth: Because the Earth goes around the Sun, our observation of distant objects such as stars results in an angular shift when observed at opposite sides of the elliptical orbit.  This shift is used as the basis of a unit knows as a parsec.  A parsec was traditionally defined as the distance where one astronomical unit subtends an angle of one arcsecond.  A parsec was redefined in 2015 to 648000/π astronomical units.  Proxima Centauri, is the nearest star to the Sun and is approximately 1.3 parsecs (4.2 light-years) from the Sun. A mega-parsec is a million parsecs.

Astronomy Mass Units

Astronomical units also apply to the mass of enormous objects such as moons, planets and stars.  For this reason, astronomy also employs mass units that compare other objects to ones familiar to us.  For example, stars are often measured in mass units of solar masses. This is a comparison of their mass to the mass of our sun (one solar_mass).   For planets, astronomers use Earth masses and Jupiter masses for understanding the relative size of rocky planets and gas giants.

Astronomy Time Units

Astronomers use the same time units as everyone else, from the very small nanoseconds, to seconds, minutes, hours, days and years.  This is true with two exceptions known as sidereal days and sidereal years.  These refer to time relative to the celestial objects (the fixed stars).  The Earth rotates every 24 hours relative to the Sun. But we are moving in a circle around the Sun.  In comparison, the Earth rotates every 23 hours, 56 minutes and 4.0905 seconds (23.9344696 hours) compared to the stars in the celestial sphere.  This is known as a sidereal day.

In the same vein, a sidereal year is the time it takes the Earth to complete one orbit around the Sun relative to the celestial sphere.  Where a year is 365 days, a sidereal year is 365.256363004 days, or 1,224.5 seconds more than a calendar year.