The Axial Rotation and Variable Macroturbulence of RR Lyrae and Red   Horizontal Branch Stars

George W. Preston; Christopher Sneden; Merieme Chadid; Ian B.; Thompson; Stephen A. Shectman

arXiv:1902.09908·astro-ph.SR·April 3, 2019

The Axial Rotation and Variable Macroturbulence of RR Lyrae and Red Horizontal Branch Stars

George W. Preston, Christopher Sneden, Merieme Chadid, Ian B., Thompson, Stephen A. Shectman

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TL;DR

This study investigates the rotational velocities and macroturbulence variations in RR Lyrae and red horizontal branch stars, revealing phase-dependent behaviors and challenging the use of surface angular momentum as a total stellar angular momentum indicator.

Contribution

The paper provides new relations between spectral line widths and velocities, and offers insights into the surface angular momentum evolution of horizontal branch stars.

Findings

01

RRab stars have an upper limit on V_macrot of 5 +/- 1 km/s.

02

RRc stars show V_macrot ranging from 2 to 12 km/s.

03

Surface angular momentum is not a reliable indicator of total stellar angular momentum.

Abstract

We have derived relations between full-width-half-maxima and equivalent widths of metallic absorption lines in the spectra of RR~Lyrae stars to estimate new upper limits on the axial equatorial rotational velocities of RR~Lyrae and metal-poor red horizontal branch stars (RHB). We also have derived the variations of RR~Lyrae macroturbulent velocities during the pulsation cycles. In RRab cycles the line widths are dominated by phase-dependent convolutions of axial rotation and macroturbulence, which we designate as V_macrot. The behavior of V_macrot is remarkably uniform among the RRab stars, but the behavior of V_macrot among RRc stars varies strongly from star to star. The RRab stars exhibit an upper limit on V_macrot of 5 +/- 1 km/s with weak evidence of an anti-correlation with T_eff. The RRc minima range from 2 to 12 km/s. The abrupt decline in large rotations with decreasing T_eff…

Tables17

Table 1. Table 1: Additional AAS T e X symbols

$≲$	`\lesssim`, `\la`	$≳$	`\gtrsim`, `\ga`
µm	`\micron`	−	`\sbond`
=	`\dbond`	≡	`\tbond`
$☉$	`\sun`	$\oplus$	`\earth`
⌀	`\diameter`
°	`\arcdeg`, `\degr`	□	`\sq`
′	`\arcmin`	″	`\arcsec`
$\overset{d}{.}$	`\fd`	$\overset{h}{.}$	`\fh`
$\overset{m}{.}$	`\fm`	$\overset{s}{.}$	`\fs`
$\overset{\circ}{.}$	`\fdg`	$\overset{'}{.}$	`\farcm`
$\overset{''}{.}$	`\farcs`	$\overset{p}{.}$	`\fp`
½	`\onehalf`	UBVR	`\ubvr`
⅓	`\onethird`	U B	`\ub`
⅔	`\twothirds`	B V	`\bv`
¼	`\onequarter`	V R	`\vr`
¾	`\threequarters`	U R	`\ur`

Table 2. Table 2: Text-mode accents

ò	`\‘{o}`	ō	`\={o}`	o͡o	`\t{oo}`
ó	`\’{o}`	ȯ	`\.{o}`	o̧	`\c{o}`
ô	`\^{o}`	ŏ	`\u{o}`	ọ	`\d{o}`
ö	`\"{o}`	ǒ	`\v{o}`	o̱	`\b{o}`
õ	`\~{o}`	ő	`\H{o}`

Table 3. Table 3: National symbols

œ	`\oe`	å	`\aa`	ł	`\l`
Œ	`\OE`	Å	`\AA`	Ł	`\L`
æ	`\ae`	ø	`\o`	ß	`\ss`
Æ	`\AE`	Ø	`\O`

Table 4. Table 4: Math-mode accents

$\hat{a}$	`\hat{a}`	$\dot{a}$	`\dot{a}`
$\overset{ˇ}{a}$	`\check{a}`	$\ddot{a}$	`\ddot{a}`
$\tilde{a}$	`\tilde{a}`	$\overset{˘}{a}$	`\breve{a}`
$\overset{´}{a}$	`\acute{a}`	$\bar{a}$	`\bar{a}`
$\overset{`}{a}$	`\grave{a}`	$\vec{a}$	`\vec{a}`

Table 5. Table 5: Greek and Hebrew letters (math mode)

$α$	`\alpha`	$ν$	`\nu`
$β$	`\beta`	$ξ$	`\xi`
$γ$	`\gamma`	$o$	`o`
$δ$	`\delta`	$π$	`\pi`
$ϵ$	`\epsilon`	$ρ$	`\rho`
$ζ$	`\zeta`	$σ$	`\sigma`
$η$	`\eta`	$τ$	`\tau`
$θ$	`\theta`	$υ$	`\upsilon`
$ι$	`\iota`	$ϕ$	`\phi`
$κ$	`\kappa`	$χ$	`\chi`
$λ$	`\lambda`	$ψ$	`\psi`
$μ$	`\mu`	$ω$	`\omega`
$ϝ$	`\digamma`	$ϰ$	`\varkappa`
$ε$	`\varepsilon`	$ς$	`\varsigma`
$ϑ$	`\vartheta`	$φ$	`\varphi`
$ϱ$	`\varrho`
$Γ$	`\Gamma`	$Σ$	`\Sigma`
$Δ$	`\Delta`	$Υ$	`\Upsilon`
$Θ$	`\Theta`	$Φ$	`\Phi`
$Λ$	`\Lambda`	$Ψ$	`\Psi`
$Ξ$	`\Xi`	$Ω$	`\Omega`
$Π$	`\Pi`
$ℵ$	`\aleph`	$ℶ$	`\beth`
$ℷ$	`\gimel`	$ℸ$	`\daleth`

Table 6. Table 6: Binary operators (math mode)

$\pm$	`\pm`	$\cap$	`\cap`
$\mp$	`\mp`	$\cup$	`\cup`
$∖$	`\setminus`	$⊎$	`\uplus`
$\cdot$	`\cdot`	$⊓$	`\sqcap`
$\times$	`\times`	$⊔$	`\sqcup`
$*$	`\ast`	$◁$	`\triangleleft`
$⋆$	`\star`	$▷$	`\triangleright`
$⋄$	`\diamond`	$≀$	`\wr`
$\circ$	`\circ`	$○$	`\bigcirc`
$∙$	`\bullet`	$△$	`\bigtriangleup`
$\div$	`\div`	$▽$	`\bigtriangledown`
$⊲$	`\lhd`	$⊳$	`\rhd`
$\lor$	`\vee`	$⊙$	`\odot`
$\land$	`\wedge`	$†$	`\dagger`
$\oplus$	`\oplus`	$‡$	`\ddagger`
$⊖$	`\ominus`	$∐$	`\amalg`
$\otimes$	`\otimes`	$⊴$	`\unlhd`
$⊘$	`\oslash`	$⊵$	`\unrhd`

Table 7. Table 7: AMS binary operators (math mode)

$∔$	`\dotplus`	$⋉$	`\ltimes`
$∖$	`\smallsetminus`	$⋊$	`\rtimes`
$⋒$	`\Cap`, `\doublecap`	$⋋$	`\leftthreetimes`
$⋓$	`\Cup`, `\doublecup`	$⋌$	`\rightthreetimes`
$⊼$	`\barwedge`	$⋏$	`\curlywedge`
$⊻$	`\veebar`	$⋎$	`\curlyvee`
$⩞$	`\doublebarwedge`
$⊟$	`\boxminus`	$⊝$	`\circleddash`
$⊠$	`\boxtimes`	$⊛$	`\circledast`
$⊡$	`\boxdot`	$⊚$	`\circledcirc`
$⊞$	`\boxplus`	$∙$	`\centerdot`
$⋇$	`\divideontimes`	$⊺$	`\intercal`

Table 8. Table 8: Miscellaneous symbols

†	`\dag`	§	`\S`
©	`\copyright`	‡	`\ddag`
¶	`\P`	£	`\pounds`
#	`\#`	$	`\$`
%	`\%`	&	`\&`
_	`\_`	{	`\{`
}	`\}`

Table 9. Table 9: Miscellaneous symbols (math mode)

$ℵ$	`\aleph`	$'$	`\prime`
$ℏ$	`\hbar`	$\emptyset$	`\emptyset`
$ı$	`\imath`	$\nabla$	`\nabla`
$ȷ$	`\jmath`	$\sqrt$	`\surd`
$ℓ$	`\ell`	$⊤$	`\top`
$℘$	`\wp`	$⊥$	`\bot`
$ℜ$	`\Re`	$∥$	`\\|`
$ℑ$	`\Im`	$∠$	`\angle`
$\partial$	`\partial`	$△$	`\triangle`
$\infty$	`\infty`	$\$	`\backslash`
$□$	`\Box`	$◇$	`\Diamond`
$\forall$	`\forall`	$♯$	`\sharp`
$\exists$	`\exists`	$♣$	`\clubsuit`
$\neg$	`\neg`	$♢$	`\diamondsuit`
$♭$	`\flat`	$♡$	`\heartsuit`
$♮$	`\natural`	$♠$	`\spadesuit`
$℧$	`\mho`

Table 10. Table 10: AMS miscellaneous symbols (math mode)

$ℏ$	`\hbar`	$‵$	`\backprime`
$ℏ$	`\hslash`	$\emptyset$	`\varnothing`
$△$	`\vartriangle`	$▲$	`\blacktriangle`
$▽$	`\triangledown`	$▼$	`\blacktriangledown`
$□$	`\square`	$■$	`\blacksquare`
$◆$	`\lozenge`	$◆$	`\blacklozenge`
$Ⓢ$	`\circledS`	$★$	`\bigstar`
$∠$	`\angle`	$∢$	`\sphericalangle`
$∡$	`\measuredangle`
$∄$	`\nexists`	$∁$	`\complement`
$℧$	`\mho`	$ð$	`\eth`
$Ⅎ$	`\Finv`	$╱$	`\diagup`
$⅁$	`\Game`	$╲$	`\diagdown`
$𝕜$	`\Bbbk`	$↾$	`\restriction`

Table 11. Table 11: Arrows (math mode)

$\leftarrow$	`\leftarrow`	$⟵$	`\longleftarrow`
$\Leftarrow$	`\Leftarrow`	$⟸$	`\Longleftarrow`
$\to$	`\rightarrow`	$⟶$	`\longrightarrow`
$\Rightarrow$	`\Rightarrow`	$⟹$	`\Longrightarrow`
$\leftrightarrow$	`\leftrightarrow`	$⟷$	`\longleftrightarrow`
$\Leftrightarrow$	`\Leftrightarrow`	$⟺$	`\Longleftrightarrow`
$\mapsto$	`\mapsto`	$⟼$	`\longmapsto`
$↩$	`\hookleftarrow`	$↪$	`\hookrightarrow`
$↼$	`\leftharpoonup`	$⇀$	`\rightharpoonup`
$↽$	`\leftharpoondown`	$⇁$	`\rightharpoondown`
$⇌$	`\rightleftharpoons`	$↝$	`\leadsto`
$↑$	`\uparrow`	$⇕$	`\Updownarrow`
$⇑$	`\Uparrow`	$↗$	`\nearrow`
$↓$	`\downarrow`	$↘$	`\searrow`
$⇓$	`\Downarrow`	$↙$	`\swarrow`
$↕$	`\updownarrow`	$↖$	`\nwarrow`

Table 12. Table 12: AMS arrows (math mode)

$⇠$	`\dashleftarrow`	$⇢$	`\dashrightarrow`
$⇇$	`\leftleftarrows`	$⇉$	`\rightrightarrows`
$⇆$	`\leftrightarrows`	$⇄$	`\rightleftarrows`
$⇚$	`\Lleftarrow`	$⇛$	`\Rrightarrow`
$↞$	`\twoheadleftarrow`	$↠$	`\twoheadrightarrow`
$↢$	`\leftarrowtail`	$↣$	`\rightarrowtail`
$↫$	`\looparrowleft`	$↬$	`\looparrowright`
$⇋$	`\leftrightharpoons`	$⇌$	`\rightleftharpoons`
$↶$	`\curvearrowleft`	$↷$	`\curvearrowright`
$↺$	`\circlearrowleft`	$↻$	`\circlearrowright`
$↰$	`\Lsh`	$↱$	`\Rsh`
$⇈$	`\upuparrows`	$⇊$	`\downdownarrows`
$↿$	`\upharpoonleft`	$↾$	`\upharpoonright`
$⇃$	`\downharpoonleft`	$⇂$	`\downharpoonright`
$⊸$	`\multimap`	$↝$	`\rightsquigarrow`
$↭$	`\leftrightsquigarrow`
$↚$	`\nleftarrow`	$↛$	`\nrightarrow`
$⇍$	`\nLeftarrow`	$⇏$	`\nRightarrow`
$↮$	`\nleftrightarrow`	$⇎$	`\nLeftrightarrow`

Table 13. Table 13: Relations (math mode)

$\leq$	`\leq`	$\geq$	`\geq`
$≺$	`\prec`	$≻$	`\succ`
$⪯$	`\preceq`	$⪰$	`\succeq`
$≪$	`\ll`	$≫$	`\gg`
$\subset$	`\subset`	$\supset$	`\supset`
$\subseteq$	`\subseteq`	$\supseteq$	`\supseteq`
$⊏$	`\sqsubset`	$⊐$	`\sqsupset`
$⊑$	`\sqsubseteq`	$⊒$	`\sqsupseteq`
$\in$	`\in`	$∋$	`\ni`
$⊢$	`\vdash`	$⊣$	`\dashv`
$⌣$	`\smile`	$∣$	`\mid`
$⌢$	`\frown`	$∥$	`\parallel`
$\neq$	`\neq`	$⟂$	`\perp`
$\equiv$	`\equiv`	$≅$	`\cong`
$\sim$	`\sim`	$⋈$	`\bowtie`
$≃$	`\simeq`	$\propto$	`\propto`
$≍$	`\asymp`	$⊧$	`\models`
$\approx$	`\approx`	$≐$	`\doteq`
		$⨝$	`\Join`

Table 14. Table 14: AMS binary relations (math mode)

$≦$	`\leqq`	$≧$	`\geqq`
$⩽$	`\leqslant`	$⩾$	`\geqslant`
$⪕$	`\eqslantless`	$⪖$	`\eqslantgtr`
$≲$	`\lesssim`	$≳$	`\gtrsim`
$⪅$	`\lessapprox`	$⪆$	`\gtrapprox`
$≊$	`\approxeq`	$≂$	`\eqsim`
$⋖$	`\lessdot`	$⋗$	`\gtrdot`
$⋘$	`\lll`, `\llless`	$⋙$	`\ggg`, `\gggtr`
$≶$	`\lessgtr`	$≷$	`\gtrless`
$⋚$	`\lesseqgtr`	$⋛$	`\gtreqless`
$⪋$	`\lesseqqgtr`	$⪌$	`\gtreqqless`
$≑$	`\doteqdot`, `\Doteq`	$≖$	`\eqcirc`
$≓$	`\risingdotseq`	$≗$	`\circeq`
$≒$	`\fallingdotseq`	$≜$	`\triangleq`
$∽$	`\backsim`	$\sim$	`\thicksim`
$≌$	`\backsimeq`	$\approx$	`\thickapprox`
$⫅$	`\subseteqq`	$⫆$	`\supseteqq`
$⋐$	`\Subset`	$⋑$	`\Supset`
$⊏$	`\sqsubset`	$⊐$	`\sqsupset`
$≼$	`\preccurlyeq`	$≽$	`\succcurlyeq`
$⋞$	`\curlyeqprec`	$⋟$	`\curlyeqsucc`
$≾$	`\precsim`	$≿$	`\succsim`
$⪷$	`\precapprox`	$⪸$	`\succapprox`
$⊲$	`\vartriangleleft`	$⊳$	`\vartriangleright`
$⊴$	`\trianglelefteq`	$⊵$	`\trianglerighteq`
$⊨$	`\vDash`	$⊩$	`\Vdash`
$⊪$	`\Vvdash`
$⌣$	`\smallsmile`	$∣$	`\shortmid`
$⌢$	`\smallfrown`	$∥$	`\shortparallel`
$≏$	`\bumpeq`	$≬$	`\between`
$≎$	`\Bumpeq`	$⋔$	`\pitchfork`
$\propto$	`\varpropto`	$϶$	`\backepsilon`
$◀$	`\blacktriangleleft`	$▶$	`\blacktriangleright`
$∴$	`\therefore`	$∵$	`\because`

Table 15. Table 15: AMS negated relations (math mode)

$≮$	`\nless`	$≯$	`\ngtr`
$≰$	`\nleq`	$≱$	`\ngeq`
$⩽̸$	`\nleqslant`	$⩾̸$	`\ngeqslant`
$≦̸$	`\nleqq`	$≧̸$	`\ngeqq`
$⪇$	`\lneq`	$⪈$	`\gneq`
$≨$	`\lneqq`	$≩$	`\gneqq`
$≨$	`\lvertneqq`	$≩$	`\gvertneqq`
$⋦$	`\lnsim`	$⋧$	`\gnsim`
$⪉$	`\lnapprox`	$⪊$	`\gnapprox`
$⊀$	`\nprec`	$⊁$	`\nsucc`
$⋠$	`\npreceq`	$⋡$	`\nsucceq`
$⪵$	`\precneqq`	$⪶$	`\succneqq`
$⋨$	`\precnsim`	$⋩$	`\succnsim`
$⪹$	`\precnapprox`	$⪺$	`\succnapprox`
$≁$	`\nsim`	$≇$	`\ncong`
$∤$	`\nshortmid`	$∦$	`\nshortparallel`
$∤$	`\nmid`	$∦$	`\nparallel`
$⊬$	`\nvdash`	$⊭$	`\nvDash`
$⊮$	`\nVdash`	$⊯$	`\nVDash`
$⋪$	`\ntriangleleft`	$⋫$	`\ntriangleright`
$⋬$	`\ntrianglelefteq`	$⋭$	`\ntrianglerighteq`
$⊈$	`\nsubseteq`	$⊉$	`\nsupseteq`
$⫅̸$	`\nsubseteqq`	$⫆̸$	`\nsupseteqq`
$⊊$	`\subsetneq`	$⊋$	`\supsetneq`
$⊊$	`\varsubsetneq`	$⊋$	`\varsupsetneq`
$⫋$	`\subsetneqq`	$⫌$	`\supsetneqq`
$⫋$	`\varsubsetneqq`	$⫌$	`\varsupsetneqq`

Table 16. Table 16: Variable-sized symbols (math mode)

$\begin{matrix} \sum & \sum & \sum & ⋂ & ⋂ & \bigcap \\ \prod & \prod & \prod & ⋃ & ⋃ & \bigcup \\ ∐ & ∐ & \coprod & ⨆ & ⨆ & \bigsqcup \\ \int & \int & \int & ⋁ & ⋁ & \bigvee \\ \oint & \oint & \oint & ⋀ & ⋀ & \bigwedge \\ ⨀ & ⨀ & \bigodot & ⨂ & ⨂ & \bigotimes \\ ⨁ & ⨁ & \bigoplus & ⨄ & ⨄ & \biguplus \end{matrix}$

Table 17. Table 17: Delimiters (math mode)

$($	`(`	$)$	`)`
$[$	`[`	$]$	`]`
${$	`\{`	$}$	`\}`
$⌊$	`\lfloor`	$⌋$	`\rfloor`
$⌈$	`\lceil`	$⌉$	`\rceil`
$⟨$	`\langle`	$⟩$	`\rangle`
$/$	`/`	$\$	`\backslash`
$\|$	`\vert`	$∥$	`\Vert`
$↑$	`\uparrow`	$⇑$	`\Uparrow`
$↓$	`\downarrow`	$⇓$	`\Downarrow`
$↕$	`\updownarrow`	$⇕$	`\Updownarrow`
$⌜$	`\ulcorner`	$⌝$	`\urcorner`
$⌞$	`\llcorner`	$⌟$	`\lrcorner`

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