`s = alpha * d`

Enter a value for all fields

The Astronomical Size (SAF) calculator uses the small angle formula (SAF) to compute the size of an astronomical body based on the distance to the object (D) and the angle subtended (α).

INSTRUCTIONS: Choose units and enter the following:

• (D) Distance to Object
• (α) Angle Subtended

Size (S): The calculator returns the size in Astronomical Units (ua).  However, this can be automatically converted to compatible units via the pull-down menu.

The Math / Science

Use the small angle formula (SAF) to compute the size of an astronomical body or separation between astronomical bodies (e.g. planet and a moon) given the distance and angle subtended:

                S = α • D

where:

• S is the Small Angle Formula size of the object.
• D is the distance to the object.
• α is the angle subtended.

This is an essential step in computing distance with the parallax method.  Note: some textbooks show this formula with a coefficient of 2.06 x 10⁵.  This is the equivalent to the formula used here, because vCalc makes the appropriate unit conversions from arc-seconds to radians and does not require the 2.06 x 10⁵ coefficient.  

Astronomy Calculators

• Hubble's Law
• Kepler's 3^rd Law formula T² = (4π • R³)/(G • M)
  • (M) - mass of the system. 
  • (T) - period of the orbit.
  • (R) - separation distance between the two objects.
  • (G) - universal gravity constant
• Small Angle Formula        Small Angle Formula    α = S / D
  • (α) - angle
  • (D) - distance to astronomical object
  • (S) - size or diameter of astronomical object
• Flux and Magnitude formulas
  • Change in Magnitude from Flux Ratio
  • Flux Ratio from Magnitudes
• Telescope formulas
  • Resolving Power of a Telescope
  • Magnification of a Telescope
• Mass formulas
  • Mass from Luminosity
  • Mass from Acceleration and Radius
  • Mass from Speed and Separation
  • Mass of extra-solar planet from mass of star, radius of star orbit and radius of exoplanet orbit
  • Mass of extra-solar planet from mass and velocity of star and velocity of planet
• Relative Size Formulas
  • Relative Size from Relative Temperature and Luminosity
  • Relative Temperature from Relative Size and Luminosity
  • Relative Luminosity from Relative Temperature and Size
• Wavelength Formulas
  • Velocity from shift in Wavelength and Unshifted Wavelength
  • Wavelength Shift from Wavelength and Speed
  • Blackbody Wavelength from Temperature
  • Blackbody Temperature from Peak Wavelength
  • Photon Energy from Wavelength
  • Photon Wavelength from Energy
• Luminosity and other Formulas
  • Distance from absolute and apparent magnitude
  • Radius from Speed and Period
  • Speed of Circular Motion
  • Hubble Time
  • Luminosity from Mass

Related Astro Calculators:

• Hubble's Law
• Kepler's 3^rd Law Calc has Kepler's 3^rd law solved for each parameter.
• Astronomy Calculator contains basic formulas for a college level Introduction to Astronomy
• Exoplanet Calculator contains formulas for studying planets outside of our Solar System.
• Astronomical Distance Calculator provides the distance from the Earth to numerous astronomical bodies (e.g. Sun, Moon, planets, stars, Milky Way's Center and Edge, Andromeda Galaxy)
• Astronomical Distance Travel Time Calculator computes the time to travel to distant parts of space at different velocities.
• Ellipse Calculator
• 3D Vector Calculator:
• Drake Equation Calculator
• Seager Equation Calculator
• Friedman Equation Calculator
• Exoplanet Calculator contains formulas for studying planets outside of our Solar System.
• Astronomical Distance Calculator provides the distance from the Earth to numerous astronomical bodies (e.g. Sun, Moon, planets, stars, Milky Way's Center and Edge, Andromeda Galaxy)
• Astronomical Distance Travel Time Calculator computes the time to travel to distant parts of space at different velocities.
• Radius of Perigee
• Radius of Apogee

Dr. Spicklemire has an instructional YouTube video on the using the Small Angle Equation (CLICK HERE).