# Is it true that we can estimate the mass of a star more accurately if it has a companion star?

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I remember I hear a theory that if a star has a companion star, we can know the mass of the star more easily, is that true? if it is true, while I understand that by measuring the period of the companion star, we can know the mass ratio of 2 stars,but how can the companion star also help to measure the mass of parent star?

In general, masses can only be precisely estimated for stars in binary systems (asteroseismology and some other indirect techniques can yield estimates too).

There are two broad types of binary where this is possible - astrometric binaries, where the motion of both stars can be seen on the sky and where the distance to the binary is known (e.g. Sirius AB); or eclipsing binaries, where the stars eclipse each other and their motions are inferred from their line of sight velocities, measured by spectroscopy (e.g. YY Gem).

You are correct, that the relative motions of the stars only yields their mass ratio. What is needed to get the individual masses is something that fixes the absolute sizes of the orbits and the orbital period, which can then be used with Kepler's third law to give the total mass of both stars and hence combined with the mass ratio to give the masses of the individual components.

In astrometric binaries you can measure the angular size of the orbits and calculate an absolute size using the known distance. In close (unresolved) binaries, that size is inferred from the velocities and orbital periods, but also requires knowledge of the inclination of the orbital plane. This is known quite accurately (is close to 90 degrees) only for eclipsing binaries.

I could present the Maths if you really like, but these are quite standard results.

I'll just fill in the blanks with the math that Rob Jeffries alluded to. There's a variety of scenarios you can consider, but I'll just run through the basics of two binary stars, where both stars are distinctly visible. Such an example is Antares, a well known and rather bright star in the northern sky. Such a star system is usually referred to as a Visual Binary.

If you want to solve for the masses of both stars, you'll need two equations. If algebra has taught you anything, it's that you need as many equations as you have unknowns if you want your system to be completely determined.

### Mass Ratio

To start with, consider two binary stars of mass $$M_1$$ and $$M_2$$ which are located at positions $$vec{r}_1$$ and $$vec{r}_2$$, respectively. The center of mass for that system, $$vec{R}$$, is defined to be

$$vec{R} = frac{vec{r}_1 M_1 + vec{r}_2 M_2}{vec{r}_1 + vec{r}_2}$$

Now, you'll notice I'm defining the positions of the stars without setting up a coordinate system. Currently $$vec{r}_1$$ and $$vec{r}_2$$ have no physical meaning. I'm going to choose a convenient coordinate system such that the origin is at the center of mass. This means that $$vec{R}=0$$ and we find that

$$frac{M_1}{M_2} = frac{vec{r}_2}{vec{r}_1}$$

This basic concept gives you the mass ratio, if only you can measure the ratio of the physical distance each star is from the center of mass. This is easily achieved by noting that a physical distance is related to the angular distance, $$alpha$$, (something quite measurable with a telescope) by the relation $$r=alpha d$$, where $$d$$ is the distance to the binary system. This allows us to say that

$$frac{M_1}{M_2} = frac{alpha_2}{alpha_1}$$

As stated, $$alpha$$ is an easily measurable parameter (more or less) by simply imaging the binary system with a telescope over a long enough time period. There's a bit more involved in this since I haven't talked at all about the inclinations of the orbits, but this is the basic premise.

### Mass Sum

Now that we have a way of getting the mass ratio, the other equation we'll look for is the sum of the masses. Anytime you hear sum of masses, you should immediately think Kepler's third law.

$$P^2 = frac{4pi^2}{G(M_1+M_2)}a^3$$

In this equation $$P$$ is the orbital period, $$G$$ is the gravitational constant, and $$a = r_1 + r_2$$ is the semi-major axis of the reduced mass system (there's some more math and assumptions behind this, but let's roll with it for now). This problem is as simple as solving for the sum of the masses. However, we'll also note that $$a = alpha d$$, where $$alpha = alpha_1 + alpha_2$$. Solving for the sum of masses gives us

$$M_1 + M_2 = frac{4pi^2}{GP^2}alpha^3d^3$$

As stated in the mass ratio section, it is a bit more involved than this because I haven't said anything about the inclination of the system. If the system has an inclination, it will tend to muck up the math and make it more difficult, but the principle concepts will be the same.

### Putting it together

You now have two equations and two unknowns. Let's solve for the masses.

$$M_1 = frac{4pi^2}{GP^2}d^3alpha^2alpha_2$$

$$M_2 = frac{4pi^2}{GP^2}d^3alpha^2alpha_1$$

## Failed Star Is One Cool Companion

Astronomers have located a planet-like star that’s barely warmer than a balmy summer day on Earth… it’s literally the coldest object ever directly imaged outside of our solar system!

WD 0806-661 B is a brown “Y dwarf” star that’s a member of a binary pair. Its companion is a much hotter white dwarf, the remains of a Sun-like star that has shed its outer layers. The pair is located about 63 light-years away, which is pretty close to us as stars go. The stars were identified by a team led by Penn State Associate Professor of Astronomy and Astrophysics Kevin Luhman using images from NASA’s Spitzer Space Telescope. Two infrared images taken in 2004 and 2009 were overlaid on top of each other and show the stars moving in tandem, indicating a shared orbit.

Of course, locating the stars wasn’t quite as easy as that. To find this stellar duo Luhman and his team searched through over six hundred images of stars located near our solar system taken years apart, looking for any shifting position as a pair.

The use of infrared imaging allowed the team to locate a dim brown dwarf star like WD 0806-661 B, which emits little visible light but shines brightly in infrared. (Even though brown dwarfs are extremely cool for stars they are still much warmer than the surrounding space. And, for the record, brown dwarfs are not actually brown.) Measurements estimate the temperature of WD 0806-661 B to be in the range of about 80 to 130 degrees Fahrenheit (26 to 54 degrees C, or 300 – 345 K)… literally body temperature!

“Essentially, what we have found is a very small star with an atmospheric temperature about cool as the Earth’s.”

– Kevin Luhman, Associate Professor of Astronomy and Astrophysics, Penn State

Six to nine times the mass of Jupiter, WD 0806-661 B is more like a planet than a star. It never accumulated enough mass to ignite thermonuclear reactions and thus more resembles a gas giant like Jupiter or Saturn. But its origins are most likely star-like, as its distance from its white dwarf companion – about 2,500 astronomical units – indicates that it developed on its own rather than forming from the other star’s disc.

There is a small chance, though, that it did form as a planet and gradually migrated out to its current distance. More research will help determine whether this may have been the case.

Brown dwarfs, first discovered in 1995, are valuable research targets because they are the next best thing to studying cool atmospheres on planets outside our solar system. Scientists keep trying to locate new record-holders for the coldest brown dwarfs, and with the discovery of WD 0806-661 B Luhman’s team has done just that!

A paper covering the team’s findings will be published in The Astrophysical Journal. Other authors of the paper include Ivo Labbé, Andrew J. Monson and Eric Persson of the Observatories of the Carnegie Institution for Science, Pasadena, Calif. Didier Saumon of the Los Alamos National Laboratory, New Mexico Mark S. Marley of the NASA Ames Research Center, Moffett Field, Calif. and John J. Bochanski also of The Pennsylvania State University.

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## North Star's Unseen Companion Photographed

Washington, DC-Light from the North Star, Polaris, has helped humans find their way for thousands of years. Yet its gravity has guided the movements of two lesser known companion stars for much longer.

One of its stellar companions is clearly visible with a telescope, but the other hugs Polaris so tightly that it has never been directly observed until now. Using the Hubble Space Telescope, astronomers have photographed this close neighbor for the first time, recording its ultraviolet light.

"The star we observed is so close to Polaris that we needed every available bit of Hubble's resolution to see it," said Nancy Evans, an astronomer at the Harvard-Smithsonian Center for Astrophysics who participated in the research.

The newly observed companion star is about 2 billion miles from Polaris. Astronomers have known about it for about 50 years from analysis of light coming from the triple star system, but it was so dim compared to Polaris that direct observation was impossible.

The observations have helped researchers refine mass estimate for the main star and the newly photographed companion.

"The companion is a little more massive than the Sun and a little brighter and a little hotter," Evans said.

Early estimates suggest that Polaris is about four times more massive than the Sun, but the researchers hope to refine their estimate with observations about the companion star's orbit.

The research was presented here today in a press conference at the 207th meeting of the American Astronomical Society.

"With Hubble, we've pulled the North Star's companion out of the shadows and into the spotlight," said astronomer Howard Bond from the Space Telescope Science Institute, which operates Hubble for NASA and the European Space Agency.

Polaris is the brightest star in the constellation Ursa Minor and located about 431 light-years away. It is situated almost directly above the celestial north pole, making it Earth's current northern polestar. It appears as a fixed point in the night sky around which all the other stars revolve, and sailors have long used it to orient themselves.

Polaris belongs to a special class of massive, pulsating stars known as Cepheids, which dim and brighten at regular intervals. Scientists use Cepheids to measure the distance to faraway galaxies and star clusters and to calculate the expansion rate of the universe. Knowing a Cepheid's mass is important to this understanding, but calculating the mass for most stars is difficult.

Although Polaris is a triple star system, it can be broken down into a binary system and a single star located farther away. Binary systems are important because their stars are among the few whose masses can be accurately determined. But calculating the mass for each star in a binary setup requires knowledge of their complete orbits. This in turn requires visual observation of their movements-something that was impossible with the Polaris binary system until now.

"Our ultimate goal is to get an accurate mass for Polaris," Evans said. "To do that, the next milestone is to measure the motion of the companion in its orbit."

## A star with a fake ID?

When we look at most stars, our observations are quite limited. We can measure stellar temperatures and colors quite easily, as well as star’s brightness. In the vast majority of cases, physical parameters like the star’s mass and radius are not directly measurable. To measure a star’s mass, we need the star to have a binary companion close enough that we can observe their orbital motion around each other. To measure a star’s radius, in general we not only need a binary companion, but the pair must be eclipsing. For a few stars we can measure their angular size directly, but this is limited to a handful of giant, nearby stars.

There are only a small percentage of stars in eclipsing binary systems, meaning a small minority of stars have measured masses and radii. In the vast majority of cases, we must rely on stellar models to estimate physical parameters. Astronomers have developed theoretical predictions for what a star of a given mass and radius (and age, metal content, rotation rate, and more!) should look like in our astronomical detectors. These models are calibrated on the few stars with known constraints.

But how good are these models? We know the models of solar-type stars are pretty good. After all, we have a really bright calibration source to compare against! Recently, there has been some evidence that the models may not be perfect: studies of M dwarfs find that their measured radii are a few percent larger than those predicted from the models. In this paper, the authors find more potential issues with the models.

From the observed white dwarf spectrum in the near-infrared, the authors determine the white dwarf is primarily made of hydrogen and has an age of nearly 8 billion years.

Matthews et al. study HD 114174, a system with a binary companion uncovered by some of the same authors last year. The binary companion is a stellar remnant called a white dwarf with a mass just over half the Sun’s. White dwarfs cool regularly and predictably, so from its luminosity and spectral energy distribution, the authors determine it must have an age of 7.77 ± 0.24 billion years. The primary star is probably very slightly larger than the Sun, with a mass of 1.05±0.05 solar masses. The authors of this paper try to measure its age via two methods. By comparing its position on an HR diagram with evolutionary models, the authors estimate the star has an age of 4.7 ± 2.5 billion years. That’s quite the uncertainty! Fortunately, they were able to apply another method: as a star ages, its rotation slows slightly as the star’s stellar wind carries away angular momentum. Thus, by measuring the star’s rotation period, its age can be estimated. (This process is called gyrochronology, and is the primary research interest of Astrobiter Ruth Angus.) For this star, the gyrochronological age is 4.0 ± 1.0 billion years.

That doesn’t make any sense! For both stars to be in a close binary system, they almost certainly formed at the same time. The white dwarf, an end state of stellar evolution, should then be slightly younger than its companion on the main sequence, yet the opposite is true. What’s going on? The authors propose a few possibilities. First, the stellar models could be incorrect, which would be a very exciting result, potentially leading to the development of new, better models. Another possibility is that the main sequence star had its rotation rate increased by some process during its life. This is plausible! Stellar lifetimes are inversely proportional to stellar masses, so the white dwarf (having reached its evolutionary end state) was originally the more massive star of the pair. As this larger binary companion went through its asymptotic giant branch phase on its way to becoming a white dwarf, it would have had a very strong stellar wind. This wind could have “spun up” the main sequence star, making its gyrochronological age appear smaller than its true age. A similar effect has been observed in another star, HD 8049.

To test this theory, the authors need more observations to improve their age estimate through evolutionary models, which is currently woefully underconstrained. With such observations, Matthews et al. will be able to determine if this star is really older than it pretends to be (maybe it’s embarrassed about its age: I don’t think there’s a Just for Stars hair dye yet!), or if our dating methods need to be improved and our physics is lacking.

## Citizen scientists discover companion star of APMPM J2036-4936

Spectrum of CWISE J2035-4936 (black) compared to the spectral standards VB 8 (M7), VB 10 (M8), LHS 2924 (M9), 2MASS 0345+2540 (L0), and 2MASS 2130-0845 (L1). Credit: Rothermich et al., 2021.

A low-mass companion to a distant star known as APMPM J2036-4936 has been recently detected as part of the citizen science project Backyard Worlds: Planet 9. The finding of the star, which received designation CWISE J203546.35-493611.0, was detailed in a paper published February 4 on the arXiv pre-print repository.

Located some 266 light years away, APMPM J2036-4936 was detected in 2005 and classified as a star of spectral type M4.5. However, subsequent observations of this object suggested that it has a spectral type M7 and an absolute G magnitude of 13.52 mag. Trying to explain this discrepancy, astronomers assume that this source may have highly unusual properties or that the wrong target was observed 16 years ago, possibly due to its large proper motion.

APMPM J2036-4936 was also imaged by NASA's Wide-ﬁeld Infrared Survey Explorer (WISE). By visually inspecting these images, Paul Beaulieu and Austin Rothermich of the Backyard Worlds: Planet 9 project have detected the presence of a low-mass companion to this star. In general, Backyard Worlds: Planet 9 is a citizen science project where volunteers examine WISE images to identify high proper motion objects.

"Here we report the discovery of CWISE J203546.35-493611.0 (hereafter CWISE J2035-4936) an object at the M/L spectral type boundary co-moving with a known M4.5 star found through the Backyard Worlds project," the scientists wrote in the paper.

Proper motion for CWISE J2035-4936 was measured to be −126 and −478 in right ascension and declination, respectively. Based on the available data, the spectral type of this star was estimated to be between M7 and L5.

Given that the spectrum of CWISE J2035-4936 does not fully match any of the spectral standards, it is difficult to draw final conclusions regarding its properties. The authors of the paper assume that the newfound object is most likely a low-mass star of spectral type M8 with low metallicity. They estimate that the companion star is separated from APMPM J2036-4936 by approximately 2,790 AU.

"When looking at the J band portion of the spectrum however, the closest ﬁt is with that of VB 8 (M8), although there are still a few minor features which do not fully match. The K band shows some slight irregularities when compared with VB 8, such as slightly lower ﬂux near the blue end. The H band of CWISE J2035-4936 however is quite peculiar, appearing to have more of a triangular shape than the H band of a standard M8, a feature that has been seen in metal low, sub-dwarf stars," the researchers explained.

They added that both CWISE J2035-4936 and APMPM J2036-4936 appear to be underluminous. Further studies of this peculiar stellar system are required to shed more light on the true nature of its two components.

## A Jupiter-mass companion to a solar-type star

The presence of a Jupiter-mass companion to the star 51 Pegasi is inferred from observations of periodic variations in the star’s radial velocity. The companion lies only about eight million kilometres from the star, which would be well inside the orbit of Mercury in our Solar System. This object might be a gas-giant planet that has migrated to this location through orbital evolution, or from the radiative stripping of a brown dwarf.

FOR more than ten years, several groups have been examining the radial velocities of dozens of stars, in an attempt to identify orbital motions induced by the presence of heavy planetary companions1 5. The precision of spectrographs optimized for Doppler studies and currently in use is limited to about 15ms"1. As the reflex motion of the Sun due to Jupiter is 13 ms1, all current searches are limited to the detection of objects with at least the mass of Jupiter (Mj). So far, all precise Doppler surveys have failed to detect any jovian planets or brown dwarfs.

## Astronomers discover most massive neutron star ever recorded

The most massive neutron star ever recorded has been discovered by astronomers 4,600 light years from Earth.

The star is more than twice the mass of the sun but just 15 miles in diameter, making it the most dense object in the universe except for black holes. It is so dense a single sugar-cube worth of neutron-star material would weigh the same as the entire human population of Earth (100 million tons).

Neutron stars are objects formed from the collapsed cores of large stars following a supernova explosion. They are also known as pulsars due to the pulses of radiation they emit as they rotate at high speeds.

Named J0740+6620, the star is 2.17 times the mass of the sun and 333,000 times the mass of the Earth, according to the paper published in Nature Astronomy. Scientists say this star is approaching the limits of how compact a single object can become without crushing in on itself.

“Neutron stars are as mysterious as they are fascinating,” sad lead researcher Thankful Cromartie, a pre-doctoral fellow at Virginia University.

“These city-sized objects are essentially ginormous atomic nuclei. They are so massive that their interiors take on weird properties. Finding the maximum mass that physics and nature will allow can teach us a great deal about this otherwise inaccessible realm in astrophysics.”

Space as seen from a grandmother's garden

### Space as seen from a grandmother's garden

The neutron star was identified by the Green Bank Telescope (GBT) in West Virginia which is so sensitive it can pick up radio waves emitted milliseconds after the birth of the universe.

“These stars are very exotic. We don’t know what they’re made of and one really important question is, ‘How massive can you make one of these stars?’ It has implications for very exotic material that we simply can’t create in a laboratory on Earth,” said Maura McLaughlin, astrophysics professor at West Virginia University.

The neutron star is a pulsar that emits beams of radio waves like a lighthouse as it spins. Pulsars get their name because of twin beams of radio waves they emit from their magnetic poles as they rotate hundreds of times each second.

Astronomers measure these radio waves to work out the mass of stellar objects. They can do this thanks to its orbiting companion star.

As the white dwarf passes in front of the pulsar star there is a subtle delay in the arrival time of the radio waves. This phenomenon – known as “Shapiro Delay” – is because the gravity from the white dwarf star slightly warps the space surrounding it.

This warping means the pulses from the rotating neutron star have to travel a little bit further. Astronomers measure that delay to calculate the mass of the white dwarf, which in turn provides a measurement of the neutron star.

Once the mass of one of the co-orbiting bodies is known they can accurately determine the mass of the other.

“The orientation of this binary star system created a fantastic cosmic laboratory,” said Scott Ransom, an astronomer at National Radio Astronomy Observatory (NRAO) and co-author on the paper.

### Recommended

“Neutron stars have this tipping point where their interior densities get so extreme that the force of gravity overwhelms even the ability of neutrons to resist further collapse.

“Each ‘most massive’ neutron star we find brings us closer to identifying that tipping point and helping us to understand the physics of matter at these mind boggling densities.”

Pulsars spin with such speed and regularity that astronomers can use them to study the nature of space-time, the mass of stellar objects and our understanding of general relativity.

## Author information

### Affiliations

Instituto de Astrofísica de Canarias, E-38205 La Laguna, Santa Cruz de Tenerife, Spain

J. Casares, A. Herrero & S. Simón-Díaz

Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Santa Cruz de Tenerife, Spain

J. Casares, A. Herrero & S. Simón-Díaz

Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain

Departament d’Astronomia i Meteorologia, Institut de Ciències del Cosmos, Universitat de Barcelona, IEEC-UB, Martí i Franquès 1, E-08028 Barcelona, Spain

Institut de Ciències de l’Espai—(IEEC-CSIC), Campus UAB, Facultat de Ciències, Torre C5 - parell - 2a planta, E-08193 Bellaterra, Spain