The Small Angle Formula calculator has three forms of the Small Angle Formula or Approximation.          Small Angle Equation

α = S / D

where:

• (α) is the angle
• (s) is the size
• (d) is the distance

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

Using the Small Angle Formula, you can compute:

• (α) - angle
• (D) - distance
• (S) - size

The Small Angle Formula is part of the free online Astronomy Calculator built by Dr. Spicklemire.

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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.

Astronomical Unit (au)	Distance from Sun (au)
149,597,870.691 kilometers   92,955,807.267 miles 499.015 light seconds     8.3169 light minutes     0.1386 light hours     0.00001581 light years 389.17 Lunar Orbit Radii	(0.39 au) Mercury (0.72 au) Venus (1.0 au) Earth (1.52 au) Mars (5.20 au) Jupiter (9.57 au) Saturn (19.16 au) Uranus (30.18 au) Neptune (39.48 au) Pluto

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!

Light Second	Light Minute	Light Hour	Light Day	Light Year	Kilo-Light Year
299,786 km 186,278 miles  0.002 au	17,987,163 km 11,176,705 miles 0.12023 au	1,079,229,797 km 670,602,305 miles 7.214 au	25,901,515,140 km 16,094,455,343 miles 173.14 au	9,460,528,405,000 km 5,878,499,814,210 miles 63,240 au 0.306 parsecs	9,460,528,405,000,000 km 5,878,499,814,210,000 miles 63,240,000 au 306 parsecs

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.

Parsec	Mega-parsec
30,856,770,000,000 kilometers 19,173,507,963,493 miles 206,264.7 au 3.26 light years	30,856,770,000,000,000,000 kilometers 19,173,507,963,493,200,000 miles 206,264,767,389 au 3,261,632 light years

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.

Earth Masses	Jupiter Masses	Solar Masses
5.97219 x1024 kg 0.00314657 Jupiter Masses 0.000003 Solar Masses	1.898 x 1027 kg 0.00095446 Solar Masses 317.806 Earth Masses	1.98855 x 1030 kilograms 1047.708 Jupiter Masses 332,968   Earth Masses

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.

Sidereal Day	Sidereal Year
23.9344696 hours	365.256363004 days

 

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Astronomy Calculators

The Astronomy Calculator Suite includes functions that are useful for studying astronomy and include the following:

• 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
  • Black Hole Mass Calculator
  • Black Hole Event Horizon Calculator
• 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
• Other Formulas
  • Distance from absolute and apparent magnitude
  • Radius from Speed and Period
  • Speed of Circular Motion
  • Hubble Time
  • Galaxy Recession Velocity
  • Luminosity from Mass
  • Hubble's Law
  • Exoplanet Calculator 
  • Orbital Radial Acceleration
  • Distance to Barycenter
  • Sakuma-Hattori Equation
  • 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.
  • 3D Vector Calculator
  • Drake Equation Calculator
  • Seager Equation Calculator
  • Friedman Equation Calculator
  • Lagrange Points L1 and L2
  • Schwarzschild Radius
  • Hill Sphere Radius
  • Escape Velocity