I have found 2 definitions of Eclipse Magnitude in total eclipse:
Solar eclipse for example, below we define $r_1$ as the radius of the Sun, $r_2$ as the radius of the Moon's Shadow, $d$ as the distance of Sun's center and Moon's center
if it is a partial eclipse, the Eclipse Magnitude have no objection:
$$ \epsilon = \frac{r_1 + r_2 - d}{2r_1} $$
Definition 1:
After the Sun is totally blocked, the Eclipse Magnitude jumps from $1$ to $r_2/r_1$, which is the ratio of Moon and Sun's diameter
http://www.jgiesen.de/eclipse/
https://www.oxfordreference.com/display/10.1093/oi/authority.20110803100126148
Definition 2:
After the Sun is totally blocked, the Eclipse Magnitude is still:
$$ \epsilon = \frac{r_1 + r_2 - d}{2r_1} > 1,\quad 0 \leq d \leq r_2 - r_1 $$
only if the Sun and Moon's center overlap, will $d = 0$, and $\epsilon = \epsilon_{max} = \dfrac{r_1 + r_2}{2r_1}$, but this will still be less than $r_2/r_1$
This provide a continuous definition of eclipse magnitude, on the other hand, in an annular solar eclipse, the Eclipse Magnitude is always $r_2/r_1$, having nothing to do with $d$, but this is continuous
https://www.geogebra.org/m/SnZ7QGTJ
Which one is right? I didn't find any books or professional articles defines the calculation of Eclipse Magnitude, Thanks!
I found this in The Explanatory Supplement to the Astronomical Almanac 3rd edition
but it do not give thecore question I think is: a total eclipse part that I want, especially lunar total eclipse, while the earth's umbra is much larger than the moon.
It said ambiguously $OA = -L_2$, but it did not say I should useThen the magnitude will be larger if the center of the Moon and the center of the Earth's umbra are closer together inSee BBottom Figure $(L_1 - m)/(L_1 + L_2)$ or
(comparing if the moon is close to the edge of the umbra inSee ETop Figure $(L_1 - L_2)/(L_1 + L_2)$) if using Definition II.
In Definition I, the magnitude will have nothing to do with center distance, as long as the moon is all in the earth's shadow.
