Just how small are X-ray binaries?

X-ray binaries are so small that we can’t directly image (i.e., spatially resolve) them, due to a combination of being small in size AND very far away.

When updating my Research page I was curious what a good analogy would be for imagining the projected size of an X-ray binary, and I’ve come up with the following. It’s just using algebra and trigonometry, so you can follow along! 😉

Let’s start with some assumptions for this little exercise: we’ll use a 10-solar-mass stellar black hole (i.e., not supermassive) that’s 2.5 kiloparsecs (~ 8000 lightyears) away. A 10-solar-mass black hole is ~2×10^{34} grams (yes, astronomers tend to use grams — cgs, as you’ll see below, means “centimeters, grams, seconds”, referring to the base units).

The radius of the black hole’s event horizon will be

R_{EH} = (2 G M) / c^2 ,

where G is the gravitational constant (6.674×10^{-8} in cgs units), M is the mass of the black hole, and c is the speed of light in a vacuum (~3×10^{10} in cgs units). Plugging these into the equation gives

R_{EH} ~ 3×10^6 cm,

or ~30 km. The distance, 2.5 kiloparsecs, is 7.7×10^{21} cm. Now we do some trigonometry to get the angular size:

A = arctan(3×10^6 cm / 7.7×10^{21} cm) = 2.2×10^{-14} degrees.

As you can see, this is a tiny, tiny number.

So let’s see how big an analogous object would be if it were on the surface of the moon. The closest distance between the surface of the earth to the surface of the moon is, approximately, 376300 km (which is the distance from center of the Earth to center of the moon, subtracted by the radius of the Earth and the radius of the moon), or 3.763×10^{10} cm.

We now want to know the size of an object that would appear to be 2.2×10^{-14} degrees in radius if it were sitting on the moon. This is

arctan(S / 3.763×10^{10} cm) = 2.2×10^{-14} degrees,

where S is the radius in centimeters. Solving this gives

S = 1.4 x 10^{-5} cm, or 0.14 micrometers in radius.

This is 1000 times smaller than the size of a single strand of human hair. Can you imagine trying to take a picture of a piece of hair that’s on the moon, let alone something 1000 times smaller? I can’t.

Let’s try something else — what about something on the surface of Mars? The smallest distance between the surface of Earth from the surface of Mars is 5.57×10^7 km, or 5.57×10^{12} cm. Using the same equation as before, but with this new distance,

arctan(S / 5.57×10^{12} cm) = 2.2 x 10^{-14} degrees,

gives S = 0.0021 cm = 0.021 mm = 21 micrometers in radius.

This is the size of a human hair (~ 30 – 100 micrometers in diameter), or one quarter of the thickness of a piece of paper!

Understandably, we don’t have instrumentation capable of imaging something this small, which is why we rely on spectral and timing measurements of photons emitted from X-ray binaries instead of just taking a picture.

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