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How do you determine the mass of a star? The most straightforward way can be done if the star is part of a binary pair. By observing the motion of two stars orbiting each other we can determine both the size of their orbits and their orbital period. From this we can determine the masses of the stars via Kepler’s laws. Binary stars provided the first measure of stellar masses, but many stars are not binary, so we must use alternative methods. 

One of the early ways of estimating the mass of a single star was through observation of its temperature and brightness. The more massive a star, the hotter and brighter it will tend to be. While this works reasonably well for main sequence stars, it isn’t overly accurate. For one thing, stars become hotter as they age, so an older star will seem somewhat more massive than it actually is.

Seismic vibration modes of the Sun. Credit: NASA/Kepler

More recently we’ve been able to use a method known as asteroseismology. It was originally used on the Sun (and then known as helioseismology). Our Sun is not a rigid object, but instead has a more fluid behavior. Solar flares and the convection from its interior create sound waves within the Sun, causing it to oscillate like a ringing bell. These oscillations can be measured by observing the motion of the Sun’s surface using Doppler measurements. Since these oscillations are affected by the density and pressure of the Sun’s interior, we can determine the mass of the Sun (among other things) accurately. We’re now able to make similar measurements of some stars, and can determine their masses in this way.

Sunspots and granules on the Sun’s surface. Credit: NASA Goddard Space Flight Center

Making accurate Doppler measurements of a star is difficult and time consuming. But it turns out we don’t need Doppler measurements to determine a star’s mass. We can simply look at the small fluctuations of a star’s brightness. While the overall brightness of most stars is fairly consistent, stars do experience small fluctuations in their brightness. This can be due to things like starspots, but it is also due to an effect known as granulation. The upper layer of a star undergoes convection, where warmer material rises to the surface pushing cooler material down. As a result, the surface of a star simmers like a pot of oatmeal. Because of this there are always small variations of a star’s brightness, which we can measure as small flickers in stellar brightness.

In a new work, astronomers compared the rate of these flickers to the mass of the star, and found that there was a good correlation between the flicker of a star and its surface gravity. Using data from the Kepler spacecraft, they were able to determine the mass of about 30,000 stars with reasonable accuracy. There are, however, limitations to this method. In particularly it only produces reasonable results for stars with a temperature between 4,500 and 7,000 Kelvin. For stars much cooler or hotter than the Sun the method has limited accuracy. However the ease of this method compared to asteroseismical methods still makes it a useful tool.

Paper: Fabienne A. Bastien, et al. A Granulation “Flicker”-based Measure of Stellar Surface Gravity.  arXiv:1512.03454 [astro-ph.SR]

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A few days ago I wrote about a planetary system discovered around an ancient star. I noted that the age of this star had been determined using asteroseismology, but how exactly does that work? What do sonic vibrations in a star have to do with its age?

The density and temperature of a Sun-like star over time. Credit: Brian E. Martin

It all has to do with the way stars age. A main sequence star such as the Sun spends most of its life in a balance between its gravitational weight trying to crush the star, and the fusion driven pressure of its core opposing that weight. While the Sun and other stars are mostly composed hydrogen, the nuclear interactions in their cores gradually transform some of that hydrogen into helium and other elements. This means that the density of a star increases with age. Since a more dense star allows gravity to squeeze it more strongly, the pressure and temperature also increase over time. Given the rate at which stars give off light energy, we can determine the rate at which fusion occurs, and thus determine how the density, pressure and temperature of a star will change over billions of years.

This is where asteroseismology comes in. The behavior of sound waves in a medium depends upon things like pressure, temperature, and density. By observing the oscillations of a star due to the sonic waves moving through it, we can categorize them into different modes. One type known as p-mode depend upon the speed of sound in the star, from which we can determine pressure and density. Another known as f-mode depends upon the surface gravity of a star. Given the density and surface gravity we can calculate the star’s mass. From these different modes we can determine where a star is along its evolutionary history, and thus determine its age.

There are other ways to determine the age of a star, and it is always useful to compare different methods for better accuracy, but one can get a fairly reasonable answer from stellar sounds alone.

Paper: Yveline Lebreton, et al. Stellar ages from asteroseismology. Proceedings IAU Symposium No. 258 (2008)

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Stars like the Sun oscillate due to acoustic waves in their interior.  The study of these sound waves in the Sun is known as helioseismology, and in other stars it’s known as asteroseismology. It turns out to be a very useful tool. While light travels slowly through the star’s interior, taking thousands of years to travel from the its core to its surface, the stellar interior is relatively transparent to acoustic waves, which means they can travel through the star at the speed of sound. Because of this, sound waves in a star can be used to study its interior, similar to the way ultrasound is used to see inside the human body. We do this by measuring the oscillations of a star’s surface using the Doppler shift of spectral lines.

Because the frequency of these acoustic waves depends upon the speed of sound, and the speed of sound depends upon the density and pressure of the star’s interior, we can use asteroseismology to determine the density of the star.  Since the density of a star depends upon its mass and temperature (and we can measure the temperature of a star by its color), asteroseismology can in principle be used to determine the size of a star very precisely.

While we have done helioseismology with the Sun for quite some time, doing asteroseismology with a star is much more difficult.  You need to measure starlight over a long period of time, and you need to analyze it in detail to determine the sound waves.  The good news is that this sort of long term data exists for some stars, such as those observed by the Kepler space telescope.  Now a new paper in the Astrophysical Journal has presents the asteroseismology results of a star known as Kepler-93.

The team used data from Kepler, and through asteroseismology determined its density to be between 1.658 and 1.646  g/cc. They also Its mass to be between 94.4% and 87.8% that of the Sun.  From this, they determined that the radius of the star is 91.9% that of the Sun, give or take 7,600 kilometers.  Basically, they measured the width of this star to within the width of the Earth, which is extraordinarily precise.

Kepler-93 has two known planets. The closer planet (Kepler-93b) transits the star about once every 5 days. From the transit data we can determine the size of the planet relative to the star, so using the transit data and the asteroseismology data the team was able to determine the size of the planet very precisely. They found that it is 1.481 times larger than Earth, give or take 120 kilometers.

Think on this for just a moment. Kepler-93 is 315 light years away, and we know the diameter of the star to within the width of the Earth. We know the diameter of one of its planets to within 120 kilometers. That’s a distance you could travel in a bit more than an hour on an interstate highway. Give or take.

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