How does a neutron star really look?

Article title: 1) The relativistic "looks" of a neutron star 2) Light Deflection Near Neutron Stars

Authors: H.P. Nollert, H. Ruder, H. Herold and U. Krauss

Institution of the first author: Lehrstuhl für Theoretische Astrophysik, Universität Tübingen, Germany

Status of Article I: open access in arXiv here Y here

Astrobite original: What does a neutron star actually look like? by Lisa Drummond

Today we will be looking at two articles that attempt to answer the question posed in the title: How do you see a neutron star up close? In the process, they illustrate (1) an interesting feature of relativity, which means that we can see the far side of a neutron star from a single point of view and (2) that this feature is relevant to understanding astrophysical observations.

In the absence of a gravitational field, the orbits of photons they are simple: they travel in a straight line! However, in the presence of a gravitational field strong, this is no longer the case. The gravitational forces of a massive body will cause the structure of the space around it to curve. The denser the object, the more the space time. The photons will follow the contours of spacetime and, therefore, will travel along curved orbits.

This is true even for stars that are not particularly dense (like our Sun). There will still be a small deviation of light that travels near the Sun due to the curvature of space-time in that region (see Figure 1). This effect was predicted by Einstein and measured successfully in 1919. This was achieved by taking photographs of a region of the sky centered on the Sun during a total solar eclipse (the solar eclipse meant that the position of the stars in that region were visible without being overshadowed by sunlight). Then, another photograph of those same stars was taken when the Sun was far from this area of the sky. The two photographs were compared and the conclusion was that the light was diverted, confirming the prediction of general relativity.

Figure 1: The Sun's gravity doubles the space-time around it. The light will continue along the fabric of curved spacetime. As the light from a background star passes through the Sun, it deviates and, therefore, the apparent position of the star is different from the actual position of the star. (Image source: http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/grel.html)

The photon orbits a neutron star

As in the case of the Sun, the gravity around a neutron star it causes spacetime to bend around it. A neutron star contains the mass of the Sun compressed into an object that is the size of a city. Therefore, relativistic effects, such as the deviation of light, will be even more pronounced near a neutron star compared to the Sun. Understanding the trajectories of photons is crucial in determining how a relativistic object, such as a star, will be of neutrons.

We usually think that space is three-dimensional, that is, it is described in terms of three coordinates, for example (X and Z). However, in General Relativity, we need four coordinates, including time, that is, (x, y, z, t). The four coordinates are closely linked; the value of the fourth coordinate (time) has an effect on the other three coordinates, therefore, we need it for a complete description of the geometry of spacetime.

For an object with spherical symmetry (which is very relevant in astrophysics, where gravitating bodies will typically have spherical geometry), the Schwarzschild metric will describe the curvature of spacetime. This metric tells us how spacetime will bend near a neutron star and, in particular, how the curvature will change as a function of the mass of the neutron star.

If we consider the equatorial movement of a photon (that is, we only care about the trajectories along a plane through the equator of the star), then the movement of the photon can be described with three coordinates, (r,Φ, t) where r and Φ are polar coordinates (see Figure 2).

Figure 2: Polar coordinates describe the geometry of an equatorial cut through a neutron star. (Image source: http://www1.kcn.ne.jp/~h-uchii/schwarzschild.html)

The equations of motion can be solved numerically to find the trajectory of a photon (that is, the angular position Φ of the photon in a particular radius r). We note that the trajectory is a function of the parameter b which is called the impact parameter. The impact parameter is the distance of the photon from a line through the center of the star (see Figure 3).

Figure 3: This figure shows the definition of Impact parameter b. The black dot corresponds to the center of the neutron star.

The value of the impact parameter b determines what type of motion will occur around the neutron star. There are two different regimes: (1) b <b_c and (2) b> b_c where b_c is the critical value of the impact parameter. Below are some examples of the two types of orbits for the two regimes in the Figure 4 Y 5, respectively. One thing we notice about the movement of photons in the vicinity of a neutron star is that not only can photons deviate as in the case of our Sun, photons can fall in orbits around the neutron star (see Figure 4).

Figure 4: Orbits of a photon around a neutron star in the b <b regime_c with b increasing from top to bottom. (Credit: Figure 2 in "Light Deflection Near Neutron Stars")

Figure 5: The photon orbits around a neutron star in the b> b regime_c, with b decreasing from top to bottom. (Credit: Figure 3 in "Light Deflection Near Neutron Stars")

Radiation of the surface of the neutron star

Due to the extreme deviation of photons around a neutron star, the appearance of a neutron star is different from physical reality. Our human minds have adapted to the gravity of the Earth and assume that the paths taken by all photons are straight. This is not a valid assumption on the surface of a neutron star, where the gravitational forces are 2 × 10^eleven times stronger than on Earth.

The radiation emitted by the neutron star is curved in such a way that parts of the back surface (normally invisible) become visible (see Figure 6). In fact, it is possible for photons to be trapped along an orbit (see Figure 4), which makes the entire surface visible. The mass of an object determines its Schwarzschild radio r_s. An object with a radius smaller than its Schwarschild radius will be a black hole. Therefore, the closer the radius of the object is to Schwarschild's radius, the more "black hole" it will be, and the more space-time it will curve around it.

Therefore, the amount that determines how much deflection a photon will experience when traveling near a neutron star is the ratio of the Schwarzschild radius to the radius of the neutron star R / r_s . As this amount decreases, more and more of the posterior surface of the neutron star is seen, as shown in the Figure 6.

Figure 6: Four images of neutron stars with the same radius R but with different masses and, therefore, different r_s. The ratio R / r_s decreases from top to bottom. (Credit: Figure 6 in "Light Deflection Near Neutron Stars")

Pulse profiles of neutron stars

Understanding the deviation of light around neutron stars is important in determining your X-ray pulse profile. This phenomenon is discussed in detail in this astrobite; The authors of today's astrobito summarize a simple model that I will briefly discuss here. In a bX-ray imaging of accretion (which is a neutron star in a binary system that increases from its non-neutron star partner), matter is channeled along the lines of the magnetic field towards the poles, where the kinetic energy becomes rays X radiation. We observe the regular pulses of this X-ray radiation as the neutron star rotates. Naturally, the radiation emitted near the poles of the neutron star will be affected by the deviation of light due to general relativity.

Figure 7: X-ray pulse profiles of an accreting X-ray binary for R / r_s = 2.4 (right column) and R / r_s = ∞ (left column). F_one corresponds to the flow of one of the polar caps and F_two corresponds to the flow of the other polar cap. F_one + F_two it is the flow of both polar caps combined. Increased deflection of the photon (ie, the decrease in R / r_s reduces modulation in the flow. (Credit: Figure 12 in "Light Deflection Near Neutron Stars")

The emission of X-rays comes from the poles of the neutron star. Without any deviation of light, both poles of the neutron star are only visible during part of the period of the neutron star and disappear at other times, therefore, the X-ray flux will be modulated throughout the period of the neutron star. neutron star. When R / r_s it is small (the neutron star is very compact), there is a substantial deviation of light and, therefore, almost the entire surface of the neutron star is visible all the time. Because both poles are visible all the time, we always see an X-ray flux. This means that there will essentially be no flux modulation when R / r_s be small, as seen in the Figure 7.

Relativity is not simply a vital building block of fundamental theoretical physics, it plays a direct role in observational astrophysics. We can use it to understand the radiation we receive from extreme and compact objects, such as neutron stars, and even learn more about the mysterious contents of their interiors.