IOAA 2015, Анализ на данни — Задача 1. Problem 1

Solution of Problem 1

a. The light curve (10 points)

Gieren 1993 (MNRAS vol 265)

b. The color curve (10 points)

Gieren 1993 (MNRAS vol 265)

c. The radial velocity curve (10 points)

Gieren 1993 (MNRAS vol 265)

Solar absolute bolometric magnitude calculation (5 points)

$F_{\odot}$ is the solar flux at the distance 10 pc which corresponds to the solar bolometric absolute magnitude.

Formula derivation (15 points)

From Stefan–Boltzmann equation we have luminosity of star:

here $R$ is radius of the star, $\sigma$ is the Stefan–Boltzmann constant, and $T$ is effective temperature of star.
Star flux at distance $d$ will be equal:

From Pogson definition we have:

Or using the Sun as reference, a star's observed bolometric flux will be:

Consider two moments, say $t_1$ and $t_2$. It is better to choose the phase $t_1$ and $t_2$ during which the star's expansion acceleration close to constant, and the difference in magnitude and color as large as possible. At the moment $t_1$, with measured temperature $T_1$ and radius $R_1$ , absolute bolometric flux will be:

.................................................................................. (1a)

Later, at moment $t_2$ :

.................................................................................. (1b)

During this time, the star's atmosphere has expanded from $R_1$ to $R_2$ :

or:

Reminding :

We have

.................................................................................. (2)

From Pogson definition, and (2) we can find:

Calculation of $\Delta R$ from the radial velocity graph (20 points)

For finding $\Delta R$, we can use radial velocity curve, taking two moments, $t_1$ and $t_2$, between which the expansion acceleration can be assumed constant,

here $P$ is pulsation period of star and $\Delta \varphi$ is phase difference in moment between $t_1$ and $t_2$.

Gieren 1993 (MNRAS vol 265)

Alternative method using integral / sum :

We can calculate the integral by drawing lines connecting two adjacent points, calculating the area of the trapesium under the line segment and sum up for all line segment between moment $t_1$ and $t_2$

So, from radial velocity curve we choose the part of the graph which is close to linear :

Which corresponds to

Now we can calculate $\Delta R$:

Temperatures and magnitudes from photometric data (10 points)

For moments $t_1$ and $t_2$ :

From light curve we have the magnitude V:

From color curve we have the color V-R

From Fig. 1 we can find the temperatures:

From table 4 we have bolometric corrections for these moments (by using Table 4 with linear interpolation):

Now we can calculate bolometric magnitudes:

Calculate $R_1$ (10 points):

First we calculate star's observed flux:

Then the radius of the star at the moment $t_1$ will be :

Calculate distance (10 points)

Answer: 9.72 kpc