Goldilocks Zone

The Goldilocks Zone calculator returns the orbital range around a star where habitable life, as we know it, is possible.  

INSTRUCTIONS: Choose units and enter the following:

• (L) Star's Luminosity
• (a) Average Planetary Albedo
• (τ) Infrared Optical Depth of Planetary Atmosphere 
  • (0 = no atmosphere; 0.8 = Earth's atmosphere)

Goldilocks Zone Range (d): The calculator returns the minimum and maximum semi-major axes (approximate radii) in astronomical units. However, this can be automatically converted to compatible distance units via the pull-down menu.

The Math / Science

The Goldilocks Zone calculator estimates the minimum and maximum distance from a star in which you can place a habitable planet.  It's based on the idea that the mean planetary temperature must be between the freezing point of water (0° C = 273 K) and the boiling point of water (100° C = 373 K).  The formula used inverts the one in the Planetary Temperature calculator to find the minimum and maximum distances.  In the formula given above, F₀ is the solar constant (1361 W/m²), the flux of light from the sun at Earth's distance from it (1 AU = 1.5x10¹¹ m).

The formula for the Goldilocks Zone is:

`d = ((L*F_0)/(4\sigma T^4)(1-a)(1+(3/4)\tau))^(1/2)`

The parameters you input are:

• d = mean distance from the center of the star (approximate semi-major axis of the orbit)
• T = the temperature.  Max distance uses (0° C = 273 K) and the min distance uses boiling point of water (100° C = 373 K).
• L = the luminosity (brightness) of the star, in units where the sun's luminosity is 1
• a =  the average planetary albedo (the fraction of light that the planet reflects in the visible region of the spectrum)
• τ =  the infrared optical depth of the planetary atmosphere (a measure of how much heat the planet traps due to the Greenhouse effect) (0 = no atmosphere; 0.8 = Earth's atmosphere)

The calculator's outputs are:

• d_min: The inner radius of the Goldilocks zone
• d_max:The outer radius of the Goldilocks zone
• W: The width of the zone

By default, the answers are given in astronomical units, but you can convert them into any units of distance you want.

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

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.