IOAA 2015 Data Analysis Problems Final SINhala&Eng SRI

1 / 8

Data Analysis Problem

Problem 1

Photometry and radial velocity data for the Cepheid type star HV2257 are given in Table 1-3, based on observations by Gieren (MNRAS vol 265, 1993) . The pulsation period of the star is 𝑃 = 39.294 days. A reference graph for the temperature – color relation and the bolometric correction tables are given in Figure 1 (Houdashelt et al., 2000) and Table 4 (http://xoomer.virgilio.it/hrtrace/Straizys.htm). Given that the solar luminosity is 𝐿⨀= 3.96 × 1026𝐽 𝑠−1 and its bolometric magnitude 𝑀⨀𝑏𝑜𝑙= 4.72. Please do

not use period-luminosity relation from the second question for this question. a. Plot the light curve based on Table 1, between phases 0.6 and 1. b. Plot the color in Table 2, between phases 0.6 and 1.

c. Plot the Radial Velocity curve from Table 3, between phases 0.6 and 1. d. Calculate the average radial velocity of the star.

e. Calculate the distance to this pulsating star using the observed data and supplementary data given in Table 4 and Figure 1. Assume that there is no extinction in this direction.

pYX~ny s[h` iAgYWsQ pYX~ñny bln~n @mhQ awr~~Qn~pwr a#wQ aAk h` iAgWYsQ vcn awhrQnñ~n. 2257 pQlQwOr s[h` phw e~ sQt bl` n#vw @m@wn~tm en~n. I @k`t@s~ iwQrQy Table 1-3 (MNRAS vol 265, 1993) in~ psE wr#vk lYEmQ@n`~sQtQ s[h` el~ sm`nyQ hwr pyQ a`r~ vr~gy sQg~m` tW @f`~ lQynE. dW qErkqW wr#@v| sY`vy ef~ sm`nyQ sQg~m` a`r~ vr~gy tW hwr @bqWm dW vr~gy lQynE. in~psE em| ek rQn em| @qk s~m`nyQ s^n @qkyQ qXm ph ef~ ek yt ef~ @qk lQy` hQr#t s`@p~k~;v wr#@v|, @b`~@l`~mQwQk sY`vy ef~ sm`nyQ ef~hQr#wQw qh@y~ bl rQn em|@b`l~ rQn em|hQr#wQw@b`l~ yt @qkqXm ph lQynE. (Houdashelt et al., 2000) pYs`rN wr#@v| nQyw w~vrNyk~ a#wQ, kl`v qXm hww~ qXm nvy pmN awr` @k`tsk, (http://xoomer.virgilio.it/hrtrace/Straizys.htm) @p`dQ tWek sh @p`dQ tW@qk k`lyn~ @qkk~ gnQmE. @p`dQ tWek s[h` nQr@p~k~; @b`@l`~mQwQk sY`vy ef~ek sm`nyQ sQg~m ar~ek vrQgy @l`kO tWek hwrbly @bqWm dW vr~gy. @p`dQ tW@qk s[h`q ef~ @qk sm`n ihw smWkrNy upsr~gy @qk @y`q` lQyn~n. 𝐿⨀= 3.96 × 1026𝐽 𝑠−1a`r~ @qk sm`nyQ a`r~ ek

{n @dl~t` a`r~, emgQn~ ef~@qk yt ef~ek t smWkrnyk~ gn~n. 𝑀⨀𝑏𝑜𝑙= 4.72@dl~t a`r~ gNny s[h` arWy pY@v|g pYs~w`r@y~ nQyw~

w~vrN @k`t@s~ wYQ@k`~n`k`r @k`t@s~ a#q vr~gply @s`yn~n. ey rQn hw qhy bl nvy mWtr a`sn~n agykQ.

a. pYs~w`r@y~ vyQ ak~;y 1 vgO@vn~ vW m#g~nQtEd| @gn h` wQrs~ ak~;y em vgO@vn~m @f~s~ kl`v @hvw~ k`ly 0.6 sQt 1 qk~v` a#wQ qw~w @gn aqQn~n.

b. 2 vgO@vn~ vW rQn a`r~ vyQ ak~;ytw~ kl`v wQrs~ a]ytw~ 0.6 sQt 1 qk~v` qw~w @gn pYs~}`rgw krnñ~n. c. 3 vgO@vn~ arWy pY@v|gy sQrs~ a]ytw~ wQrs~ ak~;y em vgO@vn~m kl`v 0.6 sQt 1 qk~v` a#wQ qw~w @gn aqQn~n. d. wr#@v| arWy pY@v|g@y~ s`m`n& agy vyQ a]@yn~ @h`~ qw~w mgQn~ @h`~ gnny krn~n

e. vgSv 4 h`1 vn r#py sh ihw nQrW]nq @y`~q` @mm s~pn~q wr#vt qEr @s`ynñ~n. sRr~yy` s[h` hQr#@g~ @b`~@l`~mQwQk vQX`lw~vy ef~ sm`nyQ el~ yt hwrk~pyQ qhyyQp`@sk~ vr~gy lQy` uw~wry wOnyQqXmwOnek qh@y~ bl rQn qhy v mW t @v`t| vlQn~ gn~n.

2 / 8

Fig. 1 The V-R color and temperature relation. Different symbols correspond to different authors. Table 1 Table 2 Table 3

Phase V mag Phase V – R Phase RadVel (km/s) 0.11 12.81 0.22 0.71 0.03 232 0.13 12.84 0.24 0.73 0.05 234 0.14 12.87 0.25 0.74 0.08 234 0.16 12.88 0.27 0.75 0.08 237 0.19 12.90 0.29 0.75 0.13 242 0.19 12.94 0.29 0.75 0.13 246 0.24 12.99 0.34 0.77 0.18 243 0.43 13.32 0.51 0.87 0.20 249 0.46 13.31 0.53 0.85 0.23 250 0.46 13.32 0.53 0.87 0.28 254 0.51 13.36 0.57 0.85 0.33 259 0.54 13.41 0.60 0.87 0.35 261 0.54 13.45 0.60 0.88 0.36 260 0.56 13.46 0.62 0.87 0.38 266 0.59 13.53 0.64 0.90 0.40 265 0.59 13.52 0.64 0.90 0.44 266 0.61 13.55 0.66 0.88 0.46 272 0.64 13.60 0.68 0.91 0.46 265 0.64 13.62 0.69 0.90 0.49 270 0.72 13.68 0.76 0.88 0.51 270

3 / 8 0.74 13.61 0.78 0.82 0.54 272 0.77 13.45 0.80 0.79 0.54 273 0.79 13.18 0.82 0.70 0.56 274 0.80 13.12 0.82 0.70 0.59 274 0.80 13.07 0.82 0.68 0.61 273 0.82 12.80 0.84 0.60 0.62 274 0.82 12.78 0.84 0.59 0.64 274 0.82 12.73 0.84 0.58 0.67 276 0.84 12.57 0.86 0.53 0.67 274 0.85 12.54 0.86 0.51 0.69 274 0.85 12.53 0.87 0.52 0.71 274 0.87 12.48 0.88 0.51 0.72 276 0.87 12.47 0.89 0.51 0.74 278 0.89 12.49 0.90 0.55 0.77 271 0.90 12.51 0.91 0.53 0.77 264 0.92 12.51 0.93 0.56 0.79 253 0.80 259 0.82 242 0.85 230 0.87 228 0.90 224 0.92 224 0.92 225 0.95 228 0.96 228

4 / 8

Table 4. Bolometric correction

T

eff

, K

BC, mag

9600

-0.25

9400

-0.16

9150

-0.10

8900

-0.03

8400

0.05

8000

0.09

7300

0.13

7100

0.11

6500

0.08

6150

0.03

5950

0.00

5800

-0.05

5500

-0.13

5250

-0.22

5050

-0.29

4950

-0.35

4850

-0.42

4700

-0.57

4600

-0.75

4400

-1.17

3900

-1.25

3750

-1.40

3550

-1.60

3400

-2.00

5 / 8

Problem 2

BVRIJHKLMN photometry of 2 stars from the constellation Cassiopeia is given in Table 5. For both stars it is believed that their light is affected by extinction by diffuse Interstellar Medium (ISM) only. Assuming that the observation is done from outside the atmosphere.

Cassiopeia w`rk` r`XQ@y~ BVRIJHKLMN pYk`X mQwQk (photometry ) qw~w Table 5 hQ q#k~@v|. em w`rk` @q@khQm a`@l`~ky extinction nQs` blp#mt lk~vn~@n~ vQsQr#nE an~wr~ w`rWy m`{&( diffuse Interstellar Medium (ISM)) mgQn~ pmNQ. sQylEm nQrWk~;N v`yE@g`~l@yn~ pQtwqW sQqE krn lq#yQ sQwn~n.

a) Using the data given in Tables 5 to 9, plot 𝐸_X−V

/𝐸

_B−V

as a function of 1/𝜆

_X

for filters B, V, R, I,

J, H, K, L, M, N for both stars. Fit approximate curves by eye (in particular, note that 𝐸X−V

/

𝐸

B−V

~ 𝑐𝑜𝑛𝑠𝑡. as 1/𝜆

X

→ 0). X is each band in the photometric system.

𝐸

B−V

is the colour excess.

Tables 5 sQt 9 qk~v` qW a#wQ qw~w mgQn~ w`rk` @qk s[h`m filters B, V, R, I, J, H, K, L, M, N s[h`

𝐸

X−V

/𝐸

B−V,

1/𝜆

X hQ XYQwyk~ @ls pYs~w`rgw krn~n. 1/𝜆X

→ 0

vQt 𝐸X−V

/𝐸

B−V

~ . nQywyk~

bv qW

a#w. pLmEv vgS wSnk~ hq~n~n pLmE vgSv ek~ek~wr#v sqh` bQs^nvW, vWQs^nvW a`r~s^nvW sQt en~s^nvW @wk~ @k`lm| qhyk~ @sù`yn~n. @qv#n~n e~a`k`rytm bQs^nvW@n`~tQ sqh`@v|.

𝐸

B−VynE v#dQmnw~ vr~NyyQ

b) Using the graphs obtained in a), estimate 𝑅_V

and 𝑅

_R

for each star.

ihw a) hQ qW gw~ pYs~w`r x`vQw@y~n~, ek~ ek~ w`rk`v s[h` 𝑅<sub>V</sub> h` 𝑅_R a#s~w@m|n~wO krn~n

𝑅

_V

=

𝐴V 𝐸B−V

and 𝑅

R

=

𝐴R 𝐸R−I

(𝐴V is the absorption in V). ( 𝐴_V ynE V hQ av@X`~;NyyQ. )

Now apply these results in order to derive a distance estimate for IC 342, a spiral galaxy in Cassiopeia obscured by Milky Way. You should assume that the properties of the ISM in IC 342 are similar to those of the ISM in our Galaxy.

q#n~ @m| lb`gw~ pYwQPl k~;Wrp}y nQs` a[Ert pw~ Cassiopeia hQ pQhQtQ sr~pQl`k`r mN~q`kQnQyk~ vn IC 342t qEr nQm`ny kQrWmt @y`q` gn~n. IC 342 hQ wQ@bn ISM vl s~vx`vy ap@g~ k~;Wrp}@y~ wQ@bn ISM vltm sm`n bv obt upkl~pny kL h#kQy.

c) Using the period-magnitude diagrams for 20 Cepheids from IC 342 (Figures 2 and 3) and assuming the period-luminosity relations:

6 / 8 IC 342 sQt @sfyQd| 20 kt a#[Q magnitude r$p (Figures 2 and 3) @y`q` gnQmQn~ h` period-luminosity sm|xn~{w` upkl~pny krmQn~ :

〈𝑀

_R

〉 = −2.91 (log (

𝑃

day

) − 1) − 4.04

and

〈𝑀

I

〉 = −3.00 (log (

𝑃

day

) − 1) − 4.06

where 〈𝑀R〉 and 〈𝑀I〉 are the mean absolute magnitudes in filters R and I, find 𝐴R for objects in

IC 342. Find the distance to IC 342.

〈𝑀_R〉 h` 〈𝑀I〉 ynE filters R h` I s[h` , m{&n& nQr@p~k~; qWp~ww` @v|. IC 342 hQ vs~wSn~ s[h` 𝐴R @s`y`, IC

342 t qEr @s`yn~n.

Table 5 BVRIJHKLMN photometry of two stars in Cassiopeia

Star MK class 𝐵 mag 𝑉 mag 𝑅 mag 𝐼 mag 𝐽 mag 𝐻 mag 𝐾 mag 𝐿 mag 𝑀 mag 𝑁 mag HD 4817 K3Iab 8.08 6.18 4.73 3.64 2.76 1.86 1.54 1.32 1.59 - HD 11092 K4II 8.66 6.57 - - 3.10 2.14 1.63 1.41 1.65 1.44

Table 6

(𝐵 − 𝑉)

₀

intrinsic colours for selected sp. types and luminosity classes

(𝐵 − 𝑉)₀ mag II Iab / Ia F0 - 0.15 G0 0.73 0.82 K0 1.06 1.18 K3 1.40 1.42 K4 1.42 1.50

7 / 8

Table 7 Infrared intrinsic colours for selected sp. types of supergiant stars (𝑉 − 𝑅)0 mag (𝑉 − 𝐼)0 mag (𝑉 − 𝐽)0 mag (𝑉 − 𝐻)0 mag (𝑉 − 𝐾)0 mag (𝑉 − 𝐿)0 mag (𝑉 − 𝑀)0 mag (𝑉 − 𝑁)0 mag F0 0.20 0.31 0.36 0.51 0.60 0.64 0.65 0.82 G0 0.55 0.90 1.14 1.52 1.71 1.72 1.72 1.98 K0 0.95 1.59 2.01 2.64 2.80 2.87 2.79 3.14 K3 1.13 1.96 2.41 3.14 3.25 3.39 3.25 3.63 K4 1.20 2.13 2.59 3.37 3.44 3.62 3.46 3.84

Table 8 Infrared intrinsic colours for selected sp. types of giant stars (𝑉 − 𝑅)0 mag (𝑉 − 𝐼)0 mag (𝑉 − 𝐽)0 mag (𝑉 − 𝐻)0 mag (𝑉 − 𝐾)0 mag (𝑉 − 𝐿)0 mag (𝑉 − 𝑀)0 mag (𝑉 − 𝑁)0 mag K0 0.60 1.03 1.23 1.72 1.94 1.97 1.90 1.92 K3 0.86 1.39 1.84 2.40 2.69 2.82 2.70 2.73 K4 0.96 1.61 2.16 2.77 3.05 3.22 3.08 3.02

Fig. 2

〈𝑅〉 is the mean apparent magnitude in filter R

〈𝑅〉

8 / 8

Table 9 Effective wavelengths of selected photometric filters

Filter B V R I J H K L M N

𝜆F/nm 450 555 670 870 1200 1620 2200 3500 5000 9000

Fig. 3

〈𝐼〉 is the mean apparent magnitude in filter I

〈𝐼〉