2018年IOAA数据分析第2题-双星系统中的致密天体 - 版本历史

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	\|(1.1±0.1)×10<sup>-12</sup>		\|(1.1±0.1)×10<sup>-12</sup>
	\|}		\|}
		+	Photometric observations indicate that the period of its light curve is identical to its orbital period, thus we may assume that the rotation period satisfies 𝑃<sub>rot</sub>=𝑃<sub>orb</sub>≡𝑃, and the inclination satisfies 𝑖<sub>orb</sub>=𝑖<sub>rot</sub>≡𝑖.
		+
		+	(D2.3.1) (16 points) '''Calculate''' the luminosity (𝐿<sub>1</sub>, in unit of 𝐿<sub>⊙</sub>), radius (𝑅<sub>1</sub>, in unit of 𝑅<sub>⊙</sub>), sine of the inclination angle (sin𝑖), as well as mass (𝑀<sub>1</sub>, in unit of 𝑀<sub>⊙</sub>) of the visible star. Please '''include''' the uncertainty in your results.
		+
		+	(D2.3.2) (4 points) '''Choose''' the correct type of this star from the following options: (1) Blue Giant (2) Yellow main sequence star (3) Red Giant (4) Red main sequence star (5) White Dwarf.
		+
		+	(D2.3.3) (10 points) Based on the mass function 𝑓(𝑀<sub>1</sub>,𝑀<sub>2</sub>) of the binary system, '''plot''' the rough relationship of 𝑀( (as vertical axis) and 𝑀<sub>1</sub> (as horizontal axis) on your graph paper and label it as Figure 5. Plot the most probable relation (by using sin𝑖), upper limit (with sin𝑖+𝛥sin𝑖) and lower limit (with sin𝑖−𝛥sin𝑖) derived in (D2.3.1).
		+
		+	(D2.3.4) (5 points) Draw a vertical '''shadowed region''' of [𝑀<sub>1</sub>−𝛥𝑀<sub>1</sub>,𝑀<sub>1</sub>+𝛥𝑀<sub>1</sub>], as well as two horizontal '''dashed lines''' showing the maximum mass of the white dwarf and neutron star, on your Figure 5. What is the possible mass of the invisible companion, and what kind of celestial object could it be?

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 !Uncertainty(km·s<sup>-1</sup>)
 |-
+|1
 |56204.9537
 | -37.417
 |0.011
 |-
+|2
 |56229.9213
 |34.846
 |0.010
 |-
+|3
 |56233.8732
 |42.567
@@ 第68行： / 第68行： @@
 F(J·s<sup>-1</sup>·m<sup>-2</sup>)
 |-
 |4890 ± 130
 |2.2 ± 0.1
 |0.272±0.049
 |14.1±0.6
 |(1.1±0.1)×10<sup>-12</sup>
 |}

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	where the subscript “1” represents the primary star and “2” represents its companion. The parameter 𝑖<sub>orb</sub> is the orbital inclination of the binary system. This mass function can also be expressed in terms of observable parameters. Calculate the '''mass function of this system''' in units of 𝑀<sub>⊙</sub>.		where the subscript “1” represents the primary star and “2” represents its companion. The parameter 𝑖<sub>orb</sub> is the orbital inclination of the binary system. This mass function can also be expressed in terms of observable parameters. Calculate the '''mass function of this system''' in units of 𝑀<sub>⊙</sub>.
		+
		+	2.3. Based on a detailed analysis on APOGEE, TRES spectra and GAIA parallax measurements, astronomers derived the following stellar parameters:
		+	{\| class="wikitable"
		+	\|+Table 5. Selected Physical Properties of 2M05215658+4359220
		+	!Effective Temperature
		+	T<sub>eff</sub>(K)
		+	!Surface Gravity
		+	log 𝑔(cm·s<sup>-2</sup>)
		+	!Parellax π
		+	（mas）
		+	!Measured Rotation
		+	Velocity
		+
		+	v<sub>rot</sub> sin i (km·s<sup>-1</sup>)
		+	!Bolometric Flux
		+	F(J·s<sup>-1</sup>·m<sup>-2</sup>)
		+	\|-
		+	\|4890 ± 130
		+	\|2.2 ± 0.1
		+	\|0.272±0.049
		+	\|14.1±0.6
		+	\|(1.1±0.1)×10<sup>-12</sup>
		+	\|}

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 .2. After discovering this peculiar star, astronomers conducted follow-up observations using the 1.5-m Tillinghast Reflector Echelle Spectrograph (TRES) at the Fred Lawrence Whipple Observatory (FLWO) located on Mt. Hopkins in Arizona, USA. The following table presents the RV measurements using this instrument:
-[[文件:2018IOAA实测第二题表格2.png|缩略图|944x944像素|2018ioaa data analysis 2.2]]
+[[文件:2018IOAA实测第二题表格2.png|缩略图|555x555px|2018ioaa data analysis 2.2]]
 (D2.2.1) (14 points) '''Plot''' the diagram of RV variation (measured with TRES) versus time on your graph paper and label it as Figure 4. Draw a suitable '''sinusoidal function''' to fit the given data. '''Estimate''' the orbital period (𝑃<sub>orb</sub>) and radial velocity semi-amplitude (𝐾) from your plot.
 (D2.2.2) (4 points) If the star is moving in a circular orbit, calculate the '''minimum value''' of the orbital radius (𝑟<sub>orb</sub>) of the star in units of both 𝑅<sub>⊙</sub> and au.
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 (D2.2.3) (7 points) The mass function of a binary system is defined as:
-$$
+<nowiki>$$
 f\left(M_{1}, M_{2}\right)=\frac{\left(M_{2} \sin i_{o r b}\right)^{3}}{\left(M_{1}+M_{2}\right)^{2}}
-$$
+$$</nowiki>
-where the subscript “1” represents the primary star and “2” represents its companion. The parameter 𝑖<sub>orb</sub> is the orbital inclination of the binary system. This mass function can also be expressed in terms of observable parameters. Calculate the mass function of this system in units of 𝑀<sub>⊙</sub>.
+where the subscript “1” represents the primary star and “2” represents its companion. The parameter 𝑖<sub>orb</sub> is the orbital inclination of the binary system. This mass function can also be expressed in terms of observable parameters. Calculate the '''mass function of this system''' in units of 𝑀<sub>⊙</sub>.

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 (D2.2.3) (7 points) The mass function of a binary system is defined as:
 $$
 f\left(M_{1}, M_{2}\right)=\frac{\left(M_{2} \sin i_{o r b}\right)^{3}}{\left(M_{1}+M_{2}\right)^{2}}
 $$

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 |+Table 3. APOGEE Radial Velocity Measurements of 2M05215658+4359220
 !Observation
 !MJD
 !RV(km·s<sup>-1</sup>)
 !Uncertainty(km·s<sup>-1</sup>)
 |-
+|
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 a_{\max }=\left.\frac{\Delta R V}{\Delta t}\right|_{\max }, \text { unit: } \mathrm{km} \mathrm{s}^{-1} \mathrm{day}^{-1}
 $$

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 |}
 (D2.1.1) (6 points) Use the data and a simple linear model to obtain an initial estimate of the apparent '''maximum acceleration''' of the star:
-<br />