We present a detailed derivation of the magnitude and obscuration formulas for solar eclipses. These quantities describe how much of the Sun’s disk is covered by the Moon — either by diameter (magnitude) or area (obscuration). We begin with basic definitions and geometry, then normalize the variables, and finally simplify the obscuration formula into a compact and practical form suitable for large-scale calculations.
Consider the Sun and Moon as perfect circles1. Define the following:
To make the math more manageable, we define normalized variables by dividing by the Sun’s radius:
\[ r = \frac{R_m}{R_s}, \quad b = \frac{d}{R_s} \]
These represent the Moon’s radius and center separation as fractions of the Sun’s radius. (We also note that \( r \) is the diameter ratio typically calculated as a fundamental quantity of interest during eclipses.)
The magnitude of an eclipse, \( M \), is the fraction of the Sun’s diameter covered by the Moon along the line connecting these two bodies' centers.
The geometric overlap length is: \[ L = R_s + R_m - d \]
Dividing by the Sun’s diameter \( 2R_s \), we obtain: \[ M = \frac{L}{2 R_s} = \frac{R_s + R_m - d}{2 R_s} \]
Substituting the normalized variables: \[ M = \frac{1 + r - b}{2} \]
Solving for \( b \) gives a useful form of the distance, preserved as an auxiliary variable: \[ \boxed{ b = r + 1 - 2M } \]
(noting that in this formula, M must be expressed as a decimal with \(M \in [0..1]\)
Magnitude gives us linear overlap, but obscuration measures the area of the Sun's disk that is blocked by the Moon — a more meaningful quantity for understanding light loss during an eclipse.
The area of overlap between two circles of radius 1 and \( r \), separated by center-to-center distance \( b \), can be derived using classical geometry. We derive the obscuration formula using a modified form of the classical area of intersection between two circles: the Sun (radius 1) and the Moon (radius r), separated by a distance b.
We begin with the classical formula for the area of overlap between two circles of radii \( R_1 \) and \( R_2 \), having center separation \( d \) [correcting the sign error in the reference]2:
\[ A = R_2^2 \cos^{-1} \left( \frac{d^2 + R_2^2 - R_1^2}{2 d R_2} \right) + R_1^2 \cos^{-1} \left( \frac{d^2 + R_1^2 - R_2^2}{2 d R_1} \right) - \frac{1}{2} \sqrt{(-d + R_2 + R_1)(d + R_2 - R_1)(d - R_2 + R_1)(d + R_2 + R_1)} \]
Set \( R_1 = R_s = 1 \) (Sun), \( R_2 = R_m/R_s = r \) (normalized Moon radius, or diameter ratio), \( b = d/R_s = d \). Then, using our defined auxiliary variables, an inverse cosine identity and algebraic manipulation, we adapt the above formula to enhance computability as follows:
\[ \text{Obsc} = \frac{1}{\pi} \left[\pi - \cos^{-1} \left( \frac{r^2 - b^2 - 1}{2b} \right) + r^2 \cos^{-1} \left( \frac{r^2 + b^2 -1}{2br} \right) + \frac{1}{2} \sqrt{(1 + r + b)(r + b - 1)(b + 1 - r)(1 + r - b)} \right] \]
(The leading division is because obscuration is a fraction of the Sun's area [\( \pi \)].) Then, define \[ s = 2br, \quad v = r^2 + b^2 - 1 \]
to simplify the inverse cosine arguments as\[ \frac{r^2 + b^2 - 1}{2br} \Rightarrow \frac{v}{s}, \quad \frac{r^2 - b^2 - 1}{2b} \Rightarrow \frac{rv - sb}{s} \]
and obtain
\[ \text{Obsc} = 1 + \frac{1}{\pi} \left[ - \cos^{-1} \left( \frac{rv - sb}{s} \right) + r^2 \cos^{-1} \left( \frac{v}{s} \right) + \frac{1}{2} \sqrt{(1 + r + b)(r + b - 1)(b + 1 - r)(1 + r - b)} \right] \]
To simplify further, observe that the pairwise products under the radical yield: \[ (1 + r + b)(r + b - 1)*(b + 1 - r)(1 + r - b) = (s + v)*(s - v) \] Thus, the radicand simplifies to \( \sqrt{s^2 - v^2} \), and the final equation is given by: \[ \boxed{ \text{Obsc} = 1 + \frac{1}{\pi} \left[ r^2 \cos^{-1} \left( \frac{v}{s} \right) - \cos^{-1} \left( \frac{rv - sb}{s} \right) + \frac{1}{2} \sqrt{s^2 - v^2} \right] } \]
This expression allows for precise numerical evaluation of the obscuration during any eclipse event, based solely on the relative apparent sizes of the Moon and Sun, and the eclipse magnitude.
Sarrvesh Seethapuram Sridhar, S. Pradeep Sundar, I. Kenny Jackson, and P. Kannan, “Detection and Analysis of Solar Eclipse”, arXiv:1206.1437 [astro-ph.IM] (2012).↩