Eclipse

The dṛkkarṇa d in yojanā-s is calculated in terms of the gnomon (R cos z), and the shadow (R sin z), as (12.1)<math display='block'> <mrow> <mi>d</mi><mo>=</mo><msqrt> <mrow> <msup> <mrow> <mrow><mo>(</mo> <mrow> <mi>D</mi><mo>&#x2212;</mo><msub> <mi>R</mi> <mi>e</mi> </msub> <mi>cos</mi><mi>z</mi> </mrow> <mo>)</mo></mrow> </mrow> <mn>2</mn> </msup> <mo>+</mo><msup> <mrow> <mrow><mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>e</mi> </msub> <mi>sin</mi><mi>z</mi> </mrow> <mo>)</mo></mrow> </mrow> <mn>2</mn> </msup> <mo>,</mo> </mrow> </msqrt> </mrow> </math>$$d = \sqrt {{{\left( {D - {R_e}\cos z} \right)}^2} + {{\left( {{R_e}\sin z} \right)}^2},}$$ where D is the dvitīya-sphuṭa-yojana-karṇa and R _e is the radius of the Earth (Refer to Figure 11.37 and equation (11.274)). The lambana-s of the Sun and Moon should be applied, to obtain their true longitudes (for the observer). When the true longitudes are the same, it is the mid-eclipse. Now, we had (12.2)<math display='block'> <mrow> <mtable columnalign='right'> <mtr columnalign='right'> <mtd columnalign='right'> <mrow> <mi>L</mi><mi>a</mi><mi>m</mi><mi>b</mi><mi>a</mi><mi>n</mi><mi>a</mi><mo>=</mo><mfrac> <mrow> <msub> <mi>R</mi> <mi>e</mi> </msub> </mrow> <mi>d</mi> </mfrac> <mo>&#x00D7;</mo><mi>D</mi><munder> <mi>r</mi> <mo>&#x02D9;</mo> </munder> <mi>g</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi> </mrow> </mtd> </mtr> <mtr columnalign='right'> <mtd columnalign='right'> <mrow> <mo>&#x2248;</mo><mfrac> <mrow> <msub> <mi>R</mi> <mi>e</mi> </msub> </mrow> <mi>D</mi> </mfrac> <mo>&#x00D7;</mo><mi>D</mi><munder> <mi>r</mi> <mo>&#x02D9;</mo> </munder> <mi>g</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math> $${\begin{array}{*{20}{r}} {Lambana = \frac{{{\operatorname{R} _e}}}{d} \times D\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{r} ggati} \\ { \approx \frac{{{\operatorname{R} _e}}}{D} \times D\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{r} ggati,} \end{array}}$$ where, we approximate d by D, the true distance from the centre of the Earth in the denominator (essentially ignoring the higher order terms in <math display='block'> <mrow> <mfrac> <mrow> <msub> <mi>R</mi> <mi>e</mi> </msub> </mrow> <mi>D</mi> </mfrac> </mrow> </math> $$\frac{{{R_e}}}{D}$$).