## Abstract

Sunspots are the most conspicuous aspects of the Sun. They have a lower temperature, as compared to the surrounding photosphere; hence, sunspots appear as dark regions on a brighter background. Sunspots cyclically appear and disappear with a 11-year periodicity and are associated with a strong magnetic field ( ∼10^{3} G) structure. Sunspots consist of a dark umbra, surrounded by a lighter penumbra. Study of umbra–penumbra area ratio can be used to give a rough idea as to how the convective energy of the Sun is transported from the interior, as the sunspot’s thermal structure is related to this convective medium.

An algorithm to extract sunspots from the white-light solar images obtained from the Kodaikanal Observatory is proposed. This algorithm computes the radius and center of the solar disk uniquely and removes the limb darkening from the image. It also separates the umbra and computes the position as well as the area of the sunspots. The estimated results are compared with the Debrecen photoheliographic results. It is shown that both area and position measurements are in quite good agreement.

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## References

Baranyi, T., Ludmany, A. 1992,

*Solar Phys.*,**139**, 247.Baranyi, T., Ludmany, A., Groyi, L.

*et al*. 1999,*ESA-SP*, 448.Cakmak, H. 2014,

*Experimental Astron.*,**37**, 539.Chapman, G. A., Walton, S. R. 2001, American Geophysical Union, Spring Meeting.

Csilla, Szasz 2003, Ph.D. Thesis.

Curto, J. J., Blanca, M., Martinez, E. 2008,

*Solar Phys.*,**250**, 411.Denker, C., Johannesson, A.,

*et al*. 1998,*Solar Phys.*,**184**, 87.Goel, S., Mathew, S. K. 2014,

*Solar Phys.*,**289**, 1413.Grossmann-Doerth, U., Schmidt, W. 1981,

*A&A*,**95**, 366.Gyori, L. 1998,

*Solar Phys.*,**180**, 109.Hiremath, K. M. 2006a, Proc. ILWS Workshop, edited by Gopalswamy, N. & Bhattacharyya, A., p. 178.

Hiremath, K. M. 2006b,

*Journal of Astron. Astrophys.*,**27**, 367.Hiremath, K. M. 2009,

*SunGe*,**4****(1)**, 16.Hiremath, K. M., Hegde, M. 2013,

*ApJ*,**763**, 137.Hiremath, K. M., Hegde, M., Soon, W. 2015,

*New Astron.*,**35**, 8.Hiremath, K. M., Lovely, M. R. 2010, preprint(2010arXiv1012.5706H).

Hiremath, K. M., Mandi, P. I. 2004,

*New Astron.*,**9****(8)**, 651.Howard, R. F. 1991,

*Solar Phys.*,**136**, 251.Howard, R. F. 1992,

*Solar Phys.*,**142**, 233.Howard, R., Gilman, P. A. Gilman, P. I. 1984,

*ApJ*,**283**, 373.Lustig, G., Wohl, H. 1995,

*Solar Phys.*,**137**, 389.Oliver, R., Ballester, J. L. 1995,

*Solar Phys.*,**165**, 145.Pettauer, T., Brandt, P. N. 1997,

*Solar Phys.*,**175**, 197.Poljancic, I., Brajsa, R., Hrzina, D.

*et al*. 2011,*Cent. Eur. Astrophyc. Bull.*,**35****(1)**, 59.Ravindra, B., Priya, T. G.

*et al*. 2013,*A&A*,**500**, A19.Singh, J., Ravindra, B. 2012,

*Bull. Astr. Soc. India*,**40**, 77.Sivaraman, K. R., Gupta, S. S. 1993,

*Solar Phys.*,**146**, 27.Smith, P. D. 1990, in:

*Practical Astronomy with your calculator*, Cambridge University Press.Steinegger, M., Brandt, P. N., Schimidt, W. 1990,

*Astonomicsche Gesellschaft Abstract Series*,**5**, 42.van Driel-Gesztelyi, Csepura, G., Nagy, I.

*et al*. 1993,*Solar Phys.*,**145**, 77.Watson, F., Fletcher, L., Dalla, S.

*et al*. 2009,*Solar Phys.*,**260**, 5.Yu, L., Deng, L., Song, F. 2014,

*Proc. 33rd Chinese Control Conference*, July 28–30.Zharkov, S., Zharkov, V., Ipson, S.

*et al*. 2005,*EURASIP Journal in Applied Signal Processing*,**15**, 2573.

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## Appendices

### Appendices

### 1.1 Appendix A

#### 1.1.1 Circle-fitting

Let *x*
_{
i
} and *y*
_{
i
} be the *x*- and *y*-coordinates of the edge detected pixels (as illustrated in Figure 2), respectively, where *i* varies from 1 to *N*, given *N* is the total number of detected pixels. Let \(\bar {x}\) and \(\bar {y}\) be the mean of the respective *x*
_{
i
} and *y*
_{
i
} coordinates. That is,

and

Firstly, we convert the ( *x*
_{
i
},*y*
_{
i
}) coordinates into a new system of coordinates ( *u*
_{
i
},*v*
_{
i
}) with

and

Let ( *u*
_{
c
},*v*
_{
c
}) be the center of the circle and let *R* be its radius in this new coordinate system. Let *α*=*R*
^{2}.

Distance of any point ( *u*
_{
i
},*v*
_{
i
}) from the center is = \(\sqrt {(u_{i} - u_{c})^{2} + (v_{i} - v_{c})^{2}}\).

According to the least square fit, best fit is obtained when the function \(S = {\sum }_{i} [g(u_{i}, v_{i})]^{2}\) is minimized, where *g*(*u*
_{
i
},*v*
_{
i
})=(*u*
_{
i
}−*u*
_{
c
})^{2}+(*v*
_{
i
}−*v*
_{
c
})^{2}−*α*. Hence, the partial derivatives of these functions with respect to *α*,*u*
_{
c
} and *v*
_{
c
} should all be zero.

*Condition* 1:

It is known that \({\sum }_{i} u_{i} = {\sum }_{i} (x_{i} - \bar {x}) = N\bar {x} - N\bar {x} = 0\). Similarly, \({\sum }_{i} v_{i} = 0\). Putting this in equation (A2), we get

*Condition* 2:

On expansion,

Substituting the value of *N*
*α* from equation (A3), the following equation is obtained:

*Condition* 3:

Proceeding the same way as in Condition 2, the following equation is obtained:

Solving simultaneous equations (A5) and (A7), the values of *u*
_{
c
} and *v*
_{
c
} are obtained. Then from equation (A3),

and

From this equation, the value of radius, *R* of the solar disc is estimated. The next step is, converting ( *u*
_{
c
},*v*
_{
c
}) into the original coordinate system, that is obtained by adding the respective mean values

and

Hence, using this method, the coordinates of the center of the image ( *x*
_{
c
},*y*
_{
c
}) and the radius *R*, is computed uniquely.

### Appendix B

### 1.1 Heliographic coordinates

Following Smith (1990), we compute the heliographic coordinates of the sunspots as follows. To compute the heliographic latitude *𝜃*, heliographic longitude *L* and longitude difference from central meridian *l*, it is necessary to calculate the daily values of heliographic latitude ( *B*
_{o}) and longitude ( *B*
_{o}) of the disk center as well as the polar angle *P*.

Let \(T = \frac {\text {JD} - 2415020}{36525}\), where JD is the Julian Date of observation and *T* is the number of Julian centuries since epoch 1900 Jan 0.5.

The geometric mean latitude \(L^{\prime }\), mean anomaly *g* and right ascension Ω of the ascending node of the Sun are

and

The true longitude *λ*
_{⊙}, of the Sun is given by

where *C* is called the equation of the center and is defined as

The apparent longitude of the Sun, *λ*
_{
a
} consists of the true longitude, *λ*
_{⊙} and corrections for aberration and nutation

The actual physical ephemeris computations begin with

The inclination of the equator of the Sun relative to the ecliptic plane is *I*=7.25^{∘} and the longitude of the ascending node of the solar equator, *K* is

*X* and *Y* are defined such that

and

where *𝜖* is the obliquity of the ecliptic and \(\lambda ^{\prime }\) is the Sun’s apparent longitude corrected for nutation.

The mean obliquity *𝜖*
_{o} is determined from

and with the correction of nutation as

Finally, polar angle *P*, *B*
_{o} and *L*
_{o} can be computed as follows:

where *M*=360^{∘}−*ϕ*. *ϕ* must be reduced to the range 0^{∘} − 360^{∘} by subtracting integral multiples of 360^{∘}.

The solar radius as viewed from the Earth changes daily due to the revolution of Earth around the Sun. Hence, the resolution of the pixels changes daily as well.

If *n* is the number of days from J2000.0, the mean anomaly *g*, measured from the epoch J2000.0 is defined as follows:

*g* is reduced to the of range 0^{∘} to 360^{∘} by adding multiples of 360^{∘}. Distance of the Sun from Earth, \(R^{\prime }\), in AU is

The semi-diameter of the Sun, Rad in arc-seconds is

Mathematical determination of the heliographic coordinates is based on the polar coordinates \((r, \theta ^{\prime })\). This means, before computation of heliographic coordinates, the observed Sun’s image in cartesian coordinates is transformed to polar coordinates. The angular distance *ρ* of any pixel from the center of the solar disc is then determined from the equation

where *R* is the radius of the solar disc as described in Appendix A, using circle fit. To calculate the heliographic latitude *𝜃* and longitude *l* from the central meridian of any pixel, the following equations are used:

The heliographic longitude is obtained by adding the value of *L*
_{o} to the the longitudinal difference *l* of the pixel from the central meridian.

For more accurate results, correction for distortion of the Sun’s image is considered. Telescope objective lens with a short focal length can contribute to distortion of the projected image. This distortion is corrected by using the following empirical relations:

and

This *ρ* is then taken as the corrected angular distance and then the heliographic coordinates are computed as mentioned above.

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Pucha, R., Hiremath, K.M. & Gurumath, S.R. Development of a Code to Analyze the Solar White-Light Images from the Kodaikanal Observatory: Detection of Sunspots, Computation of Heliographic Coordinates and Area.
*J Astrophys Astron* **37**, 3 (2016). https://doi.org/10.1007/s12036-016-9370-4

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DOI: https://doi.org/10.1007/s12036-016-9370-4