IOAA 2016, Анализ на данни — Задача 1. Двоен пулсар

D1.1

Graph Number : D1.1

• Plot uses more than 50% of graph paper: 0.5
• Axes labels ($P$ and $a$): 0.5
• Dimensions of axes: 0.5
• Ticks and values on axes (or scale written explicitly): 0.5
• Points correctly plotted:

Correctness of points: deduction of 0.5 for each wrong point.

• Errorbars on points (at least 5): 1.0

D1.2

See above

• Elliptical curve with visual best fit: 1.0
• Curve symmetric about $a = 0$ line: 0.5
• Curve symmetric about some value of $P$ ($P \approx 7588.48$): 0.5

D1.3

Values are determined from lengths of axes of ellipse and mid-point of $P$-axis.

Error margins may be determined by estimating extreme ellipses covering the points with errorbars. Any reasonable method of estimating error margins will be accepted.

$2P_{\mathrm{t}} = (1.34 \pm 0.04)\ \mathrm{\mu s}$ [$(13.4 \pm 0.4)\ \mathrm{cm}$ on graph]

$\boxed{P_{\mathrm{t}} = (0.67 \pm 0.02)\ \mathrm{\mu s}}$ [2.0]

$\boxed{P_0 = (7588.48 \pm 0.02)\ \mathrm{\mu s}}$ [2.0]

$2a_{\mathrm{t}} = (3.42 \pm 0.12)\ \mathrm{m\,s^{-2}}$ [$(17.1 \pm 0.6)\ \mathrm{cm}$ on graph] [3.0]

$\boxed{a_{\mathrm{t}} = (1.71 \pm 0.06)\ \mathrm{m\,s^{-2}}}$

• Marking table:

• Wrong values due to wrong/poor plot/fit in (D1.1) and (D1.2) WILL BE penalised.
• Error estimation is based on graph drawing. Quoted values correspond to the envelope of possible ellipses drawn to include all points with errorbars. Any reasonable method to estimate error to be given credit.

D1.4

We can easily recover the orbital period ($P_{\mathrm{B}}$) and the radius of the orbit ($r$) in a circular orbit:

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Alternative algebraic routes accepted. Each of $P_{\mathrm{B}}$ and $r$ carry 2.0 marks.

D1.5

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Errors in $P_{\mathrm{B}}$ and $r$ can be also estimated as maximum possible (worst case) error. In such case, errors would be about 1.5 times the standard error calculated above ($\delta P_{\mathrm{B}} = 0.07, \delta r = 0.25$).

D1.6

Using these newly determined orbital parameters, we can calculate the angular orbital phase for each data point, i.e., for each pair of acceleration and period measured ($P$, $a$).

Care has to be taken to choose the value of the phase from among $\phi, \pi \pm \phi, 2\pi - \phi$, depending on the sign of $\cos\phi$ and $\sin\phi$.

• Credit for each correct value: 1.0 for first three, 0.5 for last two.
• Credit for $\pi \pm \phi$ or $2\pi - \phi$ is 0.5 per value.
• All values wrong due to wrong expression for $\phi$ gets a maximum of 1.0 mark.
• Values in radians accepted.

D1.7a

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Tolerance: $\pm 0.002$ tMJD. Using $P_0$ instead of $P_{\mathrm{B}}$ gets zero.

D1.7b

where $n =$ Integer part of $[(T - T_0)/P_{\mathrm{B}}]$.

Deduction for each wrong/missing value of $n$, $T_{\mathrm{calc}}$ and $T_{\mathrm{O-C}}$: 0.5

No double penalty in one row.

D1.7c

Graph Number : D1.7

• Plot uses more than 50% of graph paper: 0.5
• Axes labels ($T_{\mathrm{O-C}}$ and $n$) including dimensions: 0.5
• Ticks and values on axes (or scale written explicitly): 0.5
• Points correctly plotted: 0.5 for each point
• Goodness of linear fit credited in next part

D1.7d

A linear fit to the plot of $T_{\mathrm{O-C}}$ vs $n$ gives the offset of period per cycle (slope) and the shift in the zero-phase point (intercept). [2.0]

This concept, which may be evident in the subsequent calculation, gains the credit, explicit statement is not necessary.

From a linear fit,

Slope $= 0.000\,43\ \mathrm{d/n} \qquad$ Intercept $= -0.010\ \mathrm{d}$ [3.0]

• Credit for good visual linear fit: 1.0
• Correct values of slope and intercept: 1.0 each
• Tolerance: $\pm 0.00002$ in slope and $\pm 0.002$ in intercept.

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Incorrect sign of correction applied carries penalty of 0.5 for each quantity.