IAO 2013, Теория, Група α — Задача 6. Supernova remnant

6.1. Age of the supernova remnant

Assume that the SNR is a sphere with the linear radius R. Using a ruler we accurately measure the diameter of the circle and the scale length (e.g. in mm) in Fig. SNR (see the figure below, the initial figure was almost 2 times larger). From these measurements we get the radius of the circle $\rho=48.5\ \mathrm{mm}$ and 100″ scale length $l=28.5\ \mathrm{mm}$. With these data we can calculate the angular radius of the SNR

If the radius R of the SNR is measured in au, the distance d in pc, and its angular radius $\theta$ in arcsec, then

The mass of the SNR is confined within the sphere of the radius R.

The expansion speed of the SNR is calculated using the expression of the kinetic energy of the SNR

As we see from the image, the darkest parts of it are distributed at distances about 60% of its radius. So we may propose that most mass of the SNR is distributed at these distances. Thus, by the above formulae we have found the average speed of matter at 0.6 R.

Assuming that the expansion velocity was constant we get the age of the SNR:

Taking into account that most data in the problem conditions was done very roughly, with maximum 1 significant digit (and even with zero significant digits as only order of magnitude is done: the energy was about $10^{46}$ J, 1% of which drives the expansion; the average density is $10^{-21}$ kg/m³), the answer should be expressed also with only one significant digit.

So the answer is: ~200 years.

Source note: the key uses $\rho$ both for the measured image radius and for the matter density, and uses $8.5\cdot10^{16}$ m in the mass calculation after $8.6\cdot10^{16}$ m above. These are retained.

6.2. Neutron-star velocity

Using a ruler we measure the distance of the neutron star from the center of circle in Fig. SNR. Say that we get $\sigma=3\ \mathrm{mm}$. Then the angular distance of the neutron star from the center of the SNR is

The linear distance of the neutron star from the center of the SNR is

The velocity of the motion of the neutron star is

Also taking into account that most data was done very roughly, the answer should be expressed also as an order of magnitude only or with only one significant digit.

The answer is: ~$10^3$ km/s.

Source note: the key prints $S_N=\sigma d$ although the substituted 10.5 is the angular distance $\delta$; its notation is preserved.

Official basic criteria

6.1. Totally 4 pt, including:

• Correct measurements of distances — 0.5 pt.
• Finding necessary distances and mass — 1.5 pt.
• Energy, finding typical speeds — 1 pt.
• Final the time, correct calculations — 1 pt.

6.2. Totally 2 pt, including:

• Correct measurements of distance from the center — 0.5 pt.
• Calculations and correct answer — 1.5 pt.

6.common. Totally 2 pt, including:

• The most data in the problem conditions was done very roughly, with maximum 1 significant digit, so the answer should be expressed also with only one significant digit, follow this principle — 2 pt.