There are times when we may want to know how bright the sun would appear to be if viewed from a different perspective, such as how bright our sun might appear if viewed from Mercury or Pluto or even from another star several light years away.
This program applies the following magnitude versus distance formula to the problem of computing the apparent ideal visual stellar magnitude of the sun as viewed from any given distance expressed in astronomical units, light years or parsecs.
NOTE:
Space is being treated as 'crystal-clear' and free of any obscuring gas or dust, which is adequate for our purposes here.
Stellar Magnitude vs. Distance
The general stellar magnitude vs. distance formula may be expressed as
Equation 1
Where:
m1 = Stellar magnitude of a star as viewed from distance d1
and
m2 = Stellar magnitude of the same star as viewed from distance d2
In this case, the astronomical body is the Sun, but it could be a another star, the moon, a planet, an asteroid, etc. Given any three known values, using Equation 1 above, we can directly derive the remaining unknown value.
When dealing with the basic problem of apparent stellar brightness and magnitude variation versus distance, there are three general units of distance we might apply, depending upon which is most convenient for the kind of work being done. These units are astronomical units (AU), light years (LY) and parsecs (pc).
First we determine how bright the Sun would appear to be if at a distance of 1.0 in each unit. That is, the magnitude of the Sun at 1.0 AU, at 1.0 light year and at 1.0 parsec. From these values we can then develop simple equations to determine how bright the Sun would appear at any distance in general for any of those units.
Let
-26.74 = Stellar magnitude of Sun at distance 1.0 AU (NASA)
1 AU = 149597870700 m (NASA)
Light Speed = 299792458 m/s (NIST)
1 Standard (Julian) Year = 365.25 days = 31557600 seconds
----------------------------------------------------
For AUs per light year, we first compute the meters
per light year and then derive AUs per light year:
1 LY = (Light Speed m/s) * (Seconds per standard Julian year)
= (299792458 m/s) * (31557600 s)
= 9460730472580800 m
Then, we can derive the number of AUs per light year from:
AU_PER_LY = (9460730472580800 m/LY) / (149597870700 m/AU)
= 63241.0770842662802687 AU/LY
-------------------------------------
For parsecs, we derive the following:
Since 1 parsec is the distance at which the sun has a parallax shift
of 1 second of arc, a parsec, expressed in AUs, equates to the number
of seconds of arc in a unit radian or about 206264.81 AUs per parsec.
ARCSEC_PER_RADIAN = ARCSEC_PER_DEG * 180 / pi
= 3600 * 180 / 3.14159265358979323846
= 206264.8062470963551565 seconds of arc per radian
AU_PER_PC = ARCSEC_PER_RADIAN
= 206264.8062470963551565 AU/pc
Now we have the numerical constants we need
for the equations in terms of AUs per unit.
1.0000000000 AU per AU
63241.0770842663 AU per LY
206264.8062470964 AU per pc
Generally, we can say that for any distance units defined in terms of AUs per distance unit: Let m = Stellar magnitude of Sun at indicated distance Then, the general equation for solar magnitude (m) becomes: m = 5 * log10(AU_PER_DISTANCE_UNIT) - 26.74 -------------------------------------------------------------- For the stellar magnitude of the sun at 1.0 astronomical unit: m = 5 * log10(AU_PER_AU) - 26.74 = 5 * log10(1) - 26.74 = 5 * 0 - 26.74 = -26.74 ------------------------------------------------------- For the stellar magnitude of the Sun at 1.0 light year: m = 5 * log10(AU_PER_LY) - 26.74 = 5 * log10(63241.0770842663) - 26.74 = 5 * 4.800999 - 26.74 = 24.004995 - 26.74 = -2.735 --------------------------------------------------- For the stellar magnitude of the Sun at 1.0 parsec: m = 5 * log10(AU_PER_PC) - 26.74 = 5 * log10(206264.8062470963) - 26.74 = 5 * 5.314425 - 26.74 = 26.572125 - 26.74 = -0.1679 ------------------------------------------------------------------ We have now derived the constants for the three distance units and can now derive simple equations specific to each unit of measure: -26.74 = Stellar magnitude of Sun at distance 1.0 AU -2.735 = Stellar magnitude of Sun at distance 1.0 LY -0.1679 = Stellar magnitude of Sun at distance 1.0 pc
The stellar magnitude versus distance equations (2, 3 and 4) were formulated specifically for application to the Sun and were derived from general Equation 1 and the computational data given above.
For any star of observed magnitude and known distance reckoned in general units:
The Derived Equations For Each Unit of Distance
For solar magnitude (m) at distance (d) reckoned in astronomical units (AU):
For solar magnitude (m) at distance (d) reckoned in light years (LY):
For solar magnitude (m) at distance (d) reckoned in parsecs (pc):
There is an obvious pattern in the design of the equations.
Equation 1
The Derived Equations For Each Unit of Distance
For solar magnitude (m) at distance (d) reckoned in astronomical units (AU):
Equation 2
For solar magnitude (m) at distance (d) reckoned in light years (LY):
Equation 3
For solar magnitude (m) at distance (d) reckoned in parsecs (pc):
Equation 4
There is an obvious pattern in the design of the equations.
References:
Solar stellar magnitude at
1.0 AU (-26.74) came from the NASA solar factsheet:http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html
Speed of light (
299792458 m/s) came from NIST:https://www.nist.gov/si-redefinition/meet-constants
Jay Tanner - 2024