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Has the amount of energy being emitted from R Hydrae's bow shock been studied?

enter image description here

If so, is there any information how much infrared radiation is emitted in any means of measurement?

If the infrared energy from the Sun, measured 1 AU from the Sun being approximately 680 watts per meter square, how would the infrared energy of the bow shock compare at the same distance measured in the same units?

Thanks to AtmosphericPrisonEscape for pointing me in the right direction. I found this table from the DETECTION OF A FAR-INFRARED BOW SHOCK NEBULA AROUND R HYA: THE FIRST MIRIAD RESULTS. enter image description here

with the CSEs being the circumstellar envelopes. Does this mean the CSE (bow shock?) is about 57% (19.7 compared to 33.9) the flux of the star at the 70 microns (mosaic) band?

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    $\begingroup$ The image IS a measurment. Probably you can find the corresponding paper by looking up the name below the image. $\endgroup$ – AtmosphericPrisonEscape Sep 23 at 15:57
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Has the amount of energy being emitted from R Hydrae's bow shock been studied?

Yes, here are a few papers:

Does this mean the CSE (bow shock?) is about 57% (19.7 compared to 33.9) the flux of the star at the 70 microns (mosaic) band?

The CSE is not a "bowshock". An envelope surrounds an object, while bowshock preceeds the direction of movement.

Reference: "The chemistry in circumstellar envelopes of evolved stars: Following the origin of the elements to the origin of life", by Lucy M. Ziurys, PNAS August 15, 2006 103 (33) 12274-12279:

"Chemically rich circumstellar shells are found around asymptotic giant branch (AGB) stars and red supergiants (RSGs). ...
Given sufficient contraction, the helium core will eventually ignite as the H shell continues to burn. The fusion of helium produces $^{12}$C, and alpha capture on $^{12}$C yields $^{16}$O in the core. Eventually, the central helium is exhausted, leaving an oxygen/carbon core surrounded by He-burning and H-burning shells. The appearance of this two-shell structure marks the beginning of the AGB phase. The primary source of luminosity in an AGB star is the H-burning shell, but periodically the He shell ignites for short periods of time. The energy released in this so-called “thermal pulse” creates a convective zone extending from shell to shell, which mixes nuclear products from the He-burning zone to the stellar surface. This process, called the “third dredge-up,” brings carbon and other elements into the detaching envelope. The thermal pulses also aid in driving the outer material from the star, creating a circumstellar shell. In fact, AGB stars lose up to 80% of their original mass in the form of an envelope, with typical rates of 10$^{−6}$ to 10$^{−4}$ M$_\odot\cdot$ yr$^{−1}$.".

A useful but rough analogy is: a plane's paint fumes are the CSE and the bowshock is caused by a difference in the speed of travel that is supersonic.

Bowshock

The image you included is used on Wikipedia's webpage "Bow shocks in astrophysics - Around other stars" and appears to have originally come from: Caltech's website, dated 2006-11-30. Another copy is on JPL's website: "Red Giant Plunging Through Space" (December 8, 2006), Spacecraft: Spitzer Space Telescope, Instrument: Multiband Imaging Photometer (MIPS). Simbad query for R Hya.

Within the webpage's text it says: "This bow shock is 16,295 astronomical units from the star to the apex and 6,188 astronomical units thick (an astronomical unit is the distance between the sun and Earth). ... The mass of the bow shock is about 400 times the mass of the Earth.".

The webpage: "About the color mapping table", which explains three of the wavelengths used in the image, says: "$\bbox[#E8D8D0, 0.15pt]{\color{yellow}{\text{Infrared}}\, 3.6 \mu m}$, $\bbox[#F8E8D0, 0.15pt]{ \color{orange}{\text{Infrared}}\, 4.5 \mu m}$, $\bbox[#E5DBD2, 0.15pt]{ \color{red}{\text{Infrared}}\, 8.0 \mu m}$". They don't explain the purple band on that webpage. The raw data is far more useful than a color mapped image prepared for public consumption.

The paper that you linked to says on page 6:

"At 165 pc (Zijlstra et al. 2002) with the -10 km s$^{-1}$ radial velocity (Knapp et al. 1998), the space velocity of R Hya is 50 $\pm$ 1 km s$^{-1}$ into the direction 12° away (receding side) from the plane of the sky. The asymmetric CO profiles (Knapp et al. 1998; Teyssier et al. 2006) can then be explained by this inclined bow shock, in which the receding and approaching shock fronts intersect with the line of sight at distinct angles.
The physical dimensions of the observed bow shock (0.03 pc thick located at 7.9 x 10$^{-2}$ ahead of the star) are consistent with the stellar-wind bow shock models for AGB stars (Wareing et al. 2006a). Adopting the mass-loss rate of 3 x 10$^{-7}$ M$_\odot$ yr$^{-1}$ (Zijlstra et al. 2002), wind velocity of 10 km s$^{-1}$ (e.g., Knapp et al. 1998), and formula for $l$ derived from the momentum conservation across the shock (Van Buren et al. 1990, eq. 2), the ambient $H$ density is 0.4 cm$^3$ and the shock surface density is 2.5 x 10$^{-7}$ g cm$^2$. Thus, the estimated amount of matter contained in the bow shock is about 1.3 x 10$^{-4}$, assuming a paraboloidal bow shock of $y = x^2 /3l$ with a 300" opening radius.".

In the paper: "Exact Analytic Solutions for Stellar Wind Bow Shocks" (March 1996), by Francis P. Wilkin, Astrophysical Journal Letters v.459, p.L31:

"Stellar wind bow shocks are cometary structures due to the supersonic passage of wind-blowing stars. They sweep up interstellar matter into thin, dense shells, which may be revealed by their postshock emission or by scattered light, and provide a means of studying winds that might otherwise go undetected. ...
The theory of momentum-supported bow shocks was first developed by Baranov, Krasnobaev, & Kulikovskii (1971, hereafter BKK), who were motivated by the problem of the interaction of the solar wind with the local interstellar medium. ...
This method allows one to obtain simple, exact solutions for all quantities in the numerical model of BKK: the shell’s shape and the mass and velocity distributions within it. The shocked fluid in the shell has momentum that is precisely the vector sum of the momenta imparted to it by the wind and ambient medium, integrated over the surface of the shell, so there is no need to divide the problem into normal and tangential parts in curvilinear coordinates. Moreover, the tail of the bow obeys a simple, momentum-conserving snowplow model in two dimensions.".

In particular, on page 3, it says:

"The tangential velocity and mass surface density are plotted in Figure 4 for several values of $a$. For most systems, $a$ is expected to be small, and the tangential velocity should prove a more sensitive way of obtaining a than the mass column.".

Figure 4
FIG. 4.—Internal shell velocity in units of V$_*$ (top) and mass surface density in units of $R_{0\rho_a}$ (bottom) for several values of $a$. Curves for $a$ = 0 and 2 are labeled, and intermediate curves correspond to $a$ = $\frac{1}{4}$, $\frac{1}{2}$, and 1. The mass per unit area $\sigma$ is defined for a line of sight that is normal to the shell. The normal direction to the shell may be obtained from eq. (6), using the relation $\hat{e}_n = \hat{e}_t$ x $\hat{e}_\phi$. If viewed from angle $\vartheta$ to the normal, the mass column $\sigma$ must be modified by a projection factor sec $\vartheta$ so long as the path length of the line of sight through the thickness of the shell is short compared to the shell’s curvature.

The exact amount of infrared radiation emitted is a function of speed and resulting temperature, it's not a means to calculate the energy from the bowshock nor is it consistent. It's a great means to visualize that there is bowshock, much as smoke or clouds help to visualize a plane's bowshock; at best it would be an indirect means to measure speed and energy.

In the monthly notices of the Royal Astronomical Society, Volume 441, Issue 1, 11 June 2014, Pages 809–812 article titled: "Detecting floating black holes as they traverse the gas disc of the Milky Way", by Xiawei Wang and Abraham Loeb they explain that bowshock can be unaccompanied by CSE and that the radiation isn't confined to the infrared bands.

"In this paper, we propose an additional observational signature of floating BHs, using the MW gas disc as a detector. As the BHs pass through the MW disc supersonically they generate a bow shock, which results in synchrotron radiation detectable at radio and infrared frequencies. ...

Observations of the Galactic disc can be used to infer $n_0$ and $T_4$. The BH speed $V_•$ can then be estimated from the Mach cone angle. The maximum Lorentz factor $\gamma_{max}$ can be inferred from the peak of the synchrotron spectrum. This, in turn, yields $B_{−5}$ based on equation (3). From the slope of the synchrotron spectrum, the power-law index $\rho$ can be estimated. Finally, with the above parameters constrained, the synchrotron flux can be used to calibrate $M_•$. The above interpretation can be verified by observing the properties of the star cluster carried by the floating BHs (O'Leary & Loeb 2009, 2012). The diffuse X-ray emission from the BH and synchrotron emission from the bow shock are supplemented by stellar emission from the star cluster around it. Since the total mass of the star cluster is much smaller than $M_•$, gravity is dominated by the BH, and thus the stars do not affect the bow shock. One can measure $M_•$ spectroscopically from the velocity dispersion of the stars as a function of distance from the BH, and verify consistency with the synchrotron flux estimate.".

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