Besselian Elements of Solar Eclipses

Derive circumstances of solar eclipses from Besselian elements

Show a map of the first solar eclipse of the year 2024:

In[1]:=

SolarEclipse[DateObject[{2024,4,8}],"EclipseMap"]

Out[1]=

Compute the Besselian element coefficients for the eclipse:

In[2]:=

elems=SolarEclipse[DateObject[{2024}],"BesselianElementsCoefficients"]

Out[2]=

ReferenceDate

Mon 8 Apr 2024 18:00:00.000TT

,ShadowAxisX{-0.318247,0.511712,0.0000331204,-8.41804×

-6

},ShadowAxisY{0.219766,0.27096,-0.0000591707,-4.65888×

-6

},ShadowAxisDeclination{0.132404,0.000259082,-2.96539×

-8

,1.76836×

-10

},ShadowAxisHourAngle{1.55861,0.261871,-1.41909×

-8

,-3.7125×

-10

},PenumbraRadius{0.535616,0.0000620222,-0.0000127553,2.06367×

-9

},UmbraRadius{-0.0105001,0.0000617134,-0.0000126917,2.05339×

-9

},PenumbraConeTangent{0.00466634,-5.28082×

-8

,-5.32346×

-10

,9.77623×

-14

},UmbraConeTangent{0.00464309,-5.25451×

-8

,-5.29694×

-10

,9.72752×

-14

}

This is the time selected as the origin of time:

In[3]:=

T0=elems["ReferenceDate"]

Out[3]=

Mon 8 Apr 2024 18:00:00.000TT

Construct the standard Besselian functions as functions of time in hours from

In[4]:=

{T0,x,y,dec,mu,l1,l2,tanf1,tanf2}=Values@SolarEclipse[DateObject[{2024}],"BesselianElementsFunctions"]

Out[4]=



Mon 8 Apr 2024 18:00:00.000TT

,Functiont.,-0.318247+0.511712t.+0.0000331204

-8.41804×

-6

,Functiont.,0.219766+0.27096t.-0.0000591707

-4.65888×

-6

,Functiont.,0.132404+0.000259082t.-2.96539×

-8

+1.76836×

-10

,Functiont.,1.55861+0.261871t.-1.41909×

-8

-3.7125×

-10

,Functiont.,0.535616+0.0000620222t.-0.0000127553

+2.06367×

-9

,Functiont.,-0.0105001+0.0000617134t.-0.0000126917

+2.05339×

-9

,Functiont.,0.00466634-5.28082×

-8

t.-5.32346×

-10

+9.77623×

-14

,Functiont.,0.00464309-5.25451×

-8

t.-5.29694×

-10

+9.72752×

-14



From these functions, compute all circumstances of the eclipse. For example, find the instant of maximum eclipse:

In[5]:=

{min,trule}=FindMinimum[Sqrt[x[t]^2+y[t]^2],{t,0}]

Out[5]=

{0.343139,{t0.308163}}

In[6]:=

T0+Quantity[t/.trule,"Hours"]

Out[6]=

Mon 8 Apr 2024 18:18:29.387TT

Compare to the reported value after changing to the

"TT"

time system:

In[7]:=

TimeSystemConvert[SolarEclipse[T0,"MaximumEclipseDate"],"TT"]

Out[7]=

Mon 8 Apr 2024 18:18:29.387TT

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Contributed by: Jeff Bryant

Besselian Elements of Solar Eclipses

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