Reference System Issues in Binary Star Calculations

Poster for Division A Meeting DAp.1.05

George H. Kaplan Consultant to U.S. Naval Observatory Washington, D.C., U.S.A. [email protected] or [email protected]

IAU General Assembly Honolulu, August 2015

1 Poster DAp.1.05

Division!A! !!!!!!Coordinate!System!Issues!in!Binary!Star!Computa3ons! George!H.!Kaplan! Consultant!to!U.S.!Naval!Observatory!!([email protected][email protected])!

It!has!been!es3mated!that!half!of!all!stars!are!components!of!binary!or!mul3ple!systems.!!Yet!the!number!of!known!orbits!for!astrometric!and!spectroscopic!binary!systems!together!is! less!than!7,000!(including!redundancies),!almost!all!of!them!for!bright!stars.!!A!new!genera3on!of!deep!allLsky!surveys!such!as!PanLSTARRS,!Gaia,!and!LSST!are!expected!to!lead!to!the! discovery!of!millions!of!new!systems.!!Although!for!many!of!these!systems,!the!orbits!may!be!poorly!sampled!ini3ally,!it!is!to!be!expected!that!combina3ons!of!new!and!old!data!sources! will!eventually!lead!to!many!more!orbits!being!known.!!As!a!result,!a!revolu3on!in!the!scien3fic!understanding!of!these!systems!may!be!upon!us.!

The!current!database!of!visual!(astrometric)!binary!orbits!represents!them!rela3ve!to!the!“plane!of!the!sky”,!that!is,!the!plane!orthogonal!to!the!line!of!sight.!!Although!the!line!of!sight! to!stars!constantly!changes!due!to!proper!mo3on,!aberra3on,!and!other!effects,!there!is!no!agreed!upon!standard!for!what!line!of!sight!defines!the!orbital!reference!plane.!! Furthermore,!the!computa3on!of!differen3al!coordinates!(component!B!rela3ve!to!A)!for!a!given!date!must!be!based!on!the!binary!system’s!direc3on!at!that!date.!!Thus,!a!different! “plane!of!the!sky”!is!appropriate!for!each!such!date,!i.e.,!each!observa3on.!!However,!projec3on!effects!between!the!reference!planes,!differen3al!aberra3on,!and!the!curvature!of!the! sky!are!generally!neglected!in!such!computa3ons.!!Usually!the!only!correc3on!applied!is!for!the!change!in!the!north!direc3on!(posi3on!angle!zero)!due!to!precession!(and!some3mes! also!proper!mo3on).!!This!paper!will!present!an!algorithm!for!a!more!complete!model!of!the!geometry!involved,!and!will!show!that!such!a!model!is!necessary!to!avoid!errors!in!the! computed!observables!that!are!significant!at!modern!astrometric!accuracy.!!The!paper!will!also!suggest!where!conven3ons!need!to!be!established!to!avoid!ambigui3es!in!how!quan33es! related!to!binary!star!orbits!are!interpreted.! !

Mo#va#on' Ambiguous'Aspects'of'the'Orbital'Parameters' Other'Geometry'to'Consider'(cont.)' This!inves3ga3on!began!as!a!rela3vely!simple!project!to!add!a! Because!ORB6!is!a!collec3on!of!orbits!determined!by!different! Another!effect!arises!from!the!3LD!nature!of!the!rela3ve!posi3ons!of! binaryLstar!so[ware!module!to!the!Naval!Observatory!Vector! inves3gators,!it!is!a!heterogeneous!source!of!informa3on.!!Several! the!two!stars.!!For!nearby,!wide!systems,!the!fact!that!B!is!closer!(or! Astrometry!So[ware!(NOVAS).!!NOVAS!it!is!a!free!sourceLcode! aspects!of!the!data!in!ORB6!are!not!well!defined!and!therefore! farther)!than!A!affects!the!separa3on!we!measure;!!we!should!not! library!for!highLprecision!astrometric!calcula3ons!available!at! present!an!obstacle!to!precise!computa3ons.!!About!1/3!of!the!orbits! assume!a!viewpoint!at!infinity!for!these!systems.!! h]p://aa.usno.navy.mil/so[ware/novas/!! do!not!have!the!epoch!E"specified!!(most!of!the!specified!E’s!are! To"Earth,"assumed"at"infinity," in!Fortran,!C,!and!Python.!!It!is!likely! 2000).!!!Orbital!elements!P!and!T,!which!are!3mes,!are!o[en! at"*me"t" that!as!observa3onal!accuracy!inL! expressed!as!a!year!+!frac3on,!in!units!of!Besselian!years,!!but!the! creases,!binary!stars!will!become! conversion!to!more!standard!3me!units!is!not!given.!!In!a!more!subtle! B" an!increasingly!important!issue!in!! issue,!it!is!unclear!what!line!of!sight!should!be!used!to!define!the! sep"2" defining!celes3al!reference!frames.!!For!example,!even!when! “plane!of!the!sky”.!!Stars!move!on!the!celes3al!sphere!(i.e.,!with! sep"2" sep"1" i’ observa3ons!span!only!a!few!years,!the!components!of!wide!pairs! respect!to!the!ICRS!axes)!due!to!proper!mo3on,!aberra3on,!and!other! δ Earth"at"*me"t" d" " with!orbital!periods!of!centuries!will!show!detectable!curvatures!of! effects.!!Aberra3on!shi[s!our!line!of!sight!to!every!star,!regardless!of! (distance"Earth"to"star"A)" A" their!paths!if!the!observa3onal!accuracy!is!great!enough!(Kaplan!&! distance,!by!41!arcseconds!over!the!course!of!every!year.!! sep 1 = (B−A) cos i’ = cos (i + Δϕ) = 70.7038 arcsec Makarov!2004,!AN!324,!419;!Makarov!&!Kaplan!2005,!AJ!129,!2420).!! ! (from previous figure)

sep 1 (B−A) cos i’ NOVAS!should!deal!with!all!types!of!astrometric!binary!systems.! To!move!forward,!I!had!to!make!several!(arbitrary)!assump3ons:! sep 2 = arctan = arctan d − δ d − (B−A) sin i ! ’ Plane"of"the" ★ If!E!is!not!specified,!then!E=2000,!i.e.,!E!=!J2000.0!=!JD!2451545.0!(TT).! sky"at"*me"t" The!binary!star!module!that!is!planned!for!NOVAS!and!is!the!subject! Again suppose B−A = 100 arcsec, i=45°, i’=45°.0555°, ★ (="*me"of" of!this!paper!will!provide!both!the!absolute!and!rela3ve!posi3ons!of! The!conversion!from!Besselian!to!Julian!epochs!and!3me!units!is!done! and d = 100 pc. ephemeris"or" observa*on)" the!two!components!of!a!binary!system,!at!a!requested!date,!given! according!to!a!conven3onal!expression!(using!fixedLlength!years)! Then sep 1 − sep 2 = 70.7038 arcsec − 70.7281 arcsec recommended!by!the!IAU!in!1976.! = −24.2 mas known!orbital!elements!and!standard!astrometric!parameters.!!Thus,! it!covers!what!may!called!the!“forward!problem”!of!producing!binary! ★ The!plane!of!the!sky!for!the!orbit!is!orthogonal!to!the!direc3on!toward! the!A!component!of!the!system!defined!by!its!mean"place"at"epoch"E.! star!ephemerides!given!an!orbit!already!determined.!!It!does!not! One!more!effect!to!consider:!differen3al!aberra3on!between!the! deal!with!the!“backward!problem”!of!determining!an!orbit!from! ★ The!direc3on!of!the!orbital!node,!i.e.,!the!sign!of!the!inclina3on,!is!! two!components.! known!(even!though!in!many!cases!it!is!ambiguous).! observa3ons.!!However,!once!an!ini3al!orbit!has!been!determined! aberrated geometric ★ Units!of!arcseconds!used!for!the!semimajor!axis!a!should!be!considered! Aberration moves the apparent places of for!a!binary!system,!followLup!observa3ons!can!improve!the!orbit! positions positions the two stars slightly differently on the to!be!a!measure!of!length;!to!convert!to!astronomical!units!(au),!divide! sky, which can affect both separations through!differen3al!correc3on;!!and!a!correct!formula3on!of!the! B" B" by!the!parallax!of!the!system.! v/c" and position angles. Differential forward!problem!is!necessary!to!avoid!systema3c!errors!in!the! projected"at"B" aberration can amount to sep"2" 1 mas for 10 arcsec separations and improved!orbital!elements.!!!That!is,!the!OLC’s!should!be!as!correct! !!!!!!!!!'''It'would'be'quite'useful'if'the'appropriate'IAU'body'would' sep"3" 10 mas for 100 arcsec separations. In this case, with sep 2 = 70.2781 arcsec, as!possible.!!Given!perfect!orbital!elements,!the!output!of!the! A" A" establish'conven#ons'such'as'the'above'for'future'binary'star'work.'' v/c" sep 2 − sep 3 could be up to ≈ 7 mas. NOVAS!module!(the!C’s)!should!be!accurate!to!be]er!than!1!mas.!!!! projected"at"A" B" ! Binary!stars!are!perhaps!the!purest!example!of!the!twoLbody! Geometry'Usually'Considered' Earth’s"orbital"velocity" problem!in!astronomy,!and!the!computa3on!of!the!rela3ve!posi3ons! v"" In!compu3ng!observables!from!orbital!elements,!we!can!take!the! A" of!the!two!components!in!a!local!spaceLfixed!reference!system!is! solu3on!of!the!twoLbody!problem!for!binary!stars!as!a!given.!! nonLcontroversial.!!However,!taking!that!informa3on!and!predic3ng!! Standard!texts!such!as!Aitken!(1964),!The"Binary"Stars,!then!advise! the!values!of!astrometric!parameters!for!an!EarthLbased!observer! correc3ng!for!the!change!in!the!north!direc3on,!due!to!precession,! requires!a!more!sophis3cated!model!of!the!whole!geometry!than!is! between!the!epoch!of!the!orbit!(E)!and!the!epoch!of!observa3on,! usually!applied.!!In!par3cular,!at!high!levels!of!astrometric!accuracy,! and!similarly!for!the!change!in!north!direc3on!due!to!the!star’s! NOVAS'Algorithm' the!3LD!nature!of!the!problem!comes!into!play!in!surprising!ways.! proper!mo3on!(although!aberra3on!is!usually!a!larger!effect).!! The!algorithm!adopted!for!the!NOVAS!double!star!algorithm!can!be! Simple!formulas!are!provided.!!Some!texts!also!give!correc3ons!for! summarized!as!follows.!!Some!details!(such!as!change!of!units)!are! change!of!scale!and!change!of!lightL3me,!for!those!systems!with! omi]ed,!and!for!prac3cal!reasons!the!order!of!computa3ons!in!the! Available'Orbital'Data'for'Binary'Stars' significant!radial!veloci3es.! so[ware!is!somewhat!different.!!! The!data!set!used!is!the!Sixth!Catalog!of!Orbits!of!Visual!Binary!Stars,! 1) Using!standard!astrometric!parameters!(e.g.,!from!the!Hipparcos!catalog)! 2 available!on!CD!from!the!U.S.!Naval!Observatory!and!online!at! Other'Geometry'To'Consider' for!component!A,!or!the!system’s!center!of!mass,!compute:! h]p://www.usno.navy.mil/USNO/astrometry/op3calLIRLprod/wds/orb6!! ''''''''''''a ,!the!mean!place!direc3on!vector!of!A!at!epoch!E"(epoch!of!orbit),! (it!is!an!update!to!Hartkopf,!Mason,!&!Worley!2001,!AJ!122,!3472).1!! In!the!module!developed!for!NOVAS,!the!change!in!north!direc3on!is! E part!of!a!general!transforma3on!between!two!3LD!reference!systems! ''''''''''''aO,!the!mean!place!direc3on!vector!of!A!at!the!3me!of!observa3on,! The!catalog!provides!2518!orbits!of!2413!systems,!although!not!all! !!!!!!!where!both!vectors!are!ss!barycentric!and!with!respect!to!the!ICRS!axes.!! systems!have!complete!sets!of!elements.!!The!elements!given!are! that!takes!into!account!the!changing!angle!at!which!we!view!the! 2) Compute!the!3x3!transforma3on!matrix!between!the!planeLofLtheLsky! the!standard!set!of!!P,!a,!e,"i,!ω,!Ω,!and!T,!where!P!is!the!period,!a!the! binary!system.!!For!wide!pairs,!the!component!of!the!stars’! coordinate!system!defined!by!a and!the!ICRS.! semimajor!axis!in!angular!measure!(usually!arcseconds),!and!T!the! separa3on!along!the!line!of!sight!(z!axis)!can!result!in!a!significant! E"" epoch!of!periastron.!!The!angular!elements!are!expressed!in! shi[!of!apparent!separa3on!when!our!viewing!angle!changes,!even! 3) Compute!the!change!in!scale!and!lightL3me!between!epoch!E!and!the! degrees,!and!P!and!T!may!be!in!a!variety!of!3me!units.!!Uncertain3es! for!the!slight!change!caused!by!aberra3on!—!see!figure!below.!!" 3me!of!observa3on,!due!to!the!radial!velocity!of!the!system.!! are!also!provided.!!The!catalog!will!be!referred!to!here!as!ORB6.! Δϕ" 4) Using!standard!Keplerian!orbit!algorithms,!applied!to!the!orbital!elements! ! given!in!ORB6,!compute!d,!the!3Lvector!displacement!of!component!B!wrt! The!reference!system!in!which!the!elements!are!expressed!is!the! component!A,!at!the!3me!of!observa3on!(corrected!for!the!change!in! “plane!of!the!sky”,!which!is!orthogonal!to!the!line!of!sight.!Within! B" lightL3me),!in!the!planeLofLtheLsky!coordinate!system!defined!by!aE.!!!! this!plane,!the!azimuthal!origin!is!the!north!direc3on!(posi3on!angle! To"Earth"at"*me"t" sep"0" 5) Correct!!d!!for!change!of!scale,!and!transform!to!the!ICRS!using!the!matrix! zero)!specified!by!an!eighth!parameter,!a!date!E,!which!represents! sep"1" from!step!2.!!!Call!the!result!d.!! i the!“equinox,!if!any,!to!which!the!node![Ω]!refers”.!!That!is,!the! 6) Using!q,!the!mass!ra3o!(if!known;!otherwise!set!q=0),!compute!!!!!!!!!!!!!!!! north!direc3on,!at!the!posi3on!of!the!binary!system!on!the!celes3al! A" sep 0 = (B−A) cos i aO"=!aO"+!q/(1+q)!d and bO "=!aO!+!1/(1+q)!d ,!the!mean!place!direc3on! sphere,!is!toward!the!north!celes3al!pole!of!date!E.!! sep 1 = (B−A) cos (i + Δϕ) vectors!of!components!A!and!B,!at!the!3me!of!observa3on,!in!the!ICRS.! ! Suppose B−A = 100 arcsec, i=45°, and 7) Compute!the!apparent!place!direc3on!vectors!separately!for!A!and!B! ! Δϕ = 20 arcsec = 0.00555°. components,!star3ng!with!aO!and!bO ;!!apply!parallax,!gravita3onal!light! ! Plane"of"the" 1! Then sep 0 − sep 1 = 70.7107 arcsec bending,!and!aberra3on.!!Obtain!apparent!α!and!δ!!for!both!components.! The!orbital!elements!of!spectroscopicLonly!binary!systems,!for!example,!those!listed!in!the! − 70.7038 arcsec = 6.9 mas sky"at"*me"t""""" Plane"of"the" SB9!catalog,!are!not!suitable!for!this!project!since!Ω!is!not!known,!and!a!and!i!are!known! (="*me"of" sky"at"*me"E" 8) Compute!differen3al!apparent!coordinates!(posi3on!angle!θ!!and! only!in!the!combina3on!a"sin"i.! Position angles could also be affected, ephemeris"or" (="orbit’s"equinox)" depending on the relative orientations observa*on)" separa3on!ρ)!between!A!and!B." of the planes. What is this About?

• Computing ephemerides of visual (astrometric) binary stars with known orbits

• Ephemerides consist of values of astrometric observables for the pair of stars, for a specific time:  Position angle (θ) and separation (ρ)  Offsets in RA and Dec: Δα and Δδ  “Absolute” apparent RA and Dec: α and δ (in some standard reference system) • It’s not about computing orbits from observations • It assumes there is one set of astrometric parameters

(α, δ, µα, µδ, π, vR) for the pair, which might apply to either the A component or center of mass

3 Motivation

• Add new software module Naval Observatory Vector Astrometry Software (NOVAS)

• NOVAS it is a free source-code library for high-precision astrometric calculations available at

http://aa.usno.navy.mil/software/novas/

in Fortran, C, and Python.

• It is likely that as observational accuracy increases, binary stars will become an increasingly important issue in defining celestial reference frames

NOVAS has to accommodate them!

4 Orbit Data to Work With

• Sixth Catalog of Orbits of Visual Binary Stars (ORB6) http://www.usno.navy.mil/USNO/astrometry/optical-IR-prod/wds/orb6 Update to Hartkopf, Mason, & Worley 2001, AJ 122, 3472

• 2518 orbits for 2413 systems

 Some systems have more than one orbit

 Not all systems have complete sets of elements

 Most, but not all, are Hipparcos stars

• Elements are standard set of P, a, e, i, ω, Ω, T

 Plus uncertainties, references, and other information

• Also epoch E: equinox for Ω  Pole of date for north reference, i.e., position angle zero 5 Some Things I Wish Were Specified

• Epoch E not given for 33% of orbits

• Besselian years are frequently used for periods and periastron date −− what conversion factor to “normal” time units was used?

• What direction defines the “plane of the sky” −− the plane orthogonal to the line of sight?

6 Assumptions I Had to Make About the Orbits (whether valid or not) • If E is not specified, then E=2000, i.e., E = J2000.0 = JD 2451545.0 (TT). • The conversion from Besselian to Julian epochs and time units is done according to a conventional expression (using fixed- length years) recommended by the IAU in 1976. • The plane of the sky for the orbit is orthogonal to the direction toward the A component of the system defined by its mean place at epoch E. • The direction of the orbital node, i.e., the sign of the inclination, is known (even though in many cases it is ambiguous). • Units of arcseconds used for the semimajor axis a should be considered to be a measure of length; to convert to astronomical units (au), divide by the parallax of the system.

7 Conventional Approach to Binary Star Ephemerides

• Plane of sky doesn’t change

• System is viewed from infinity

• Standard 2-body code works fine for the relative positions of the components in their local reference system

• Once that is done, there are no geometric or light- propagation effects that significantly shift the apparent position of one component relative to the other

• Do need to adjust for the change in north direction between epoch E of the orbit and date of observation (i.e., date of ephemerides)

8 My Approach to Binary Star Ephemerides for NOVAS

• Plane of sky continually changes, i.e., our line of sight to the system changes and should be accounted for

 Done via a 3-D transformation of one coordinate system to another (which automatically adjusts for the change in north direction)

• System must be viewed from its correct distance  Parallaxes are known for most systems

• Standard 2-body code works fine for the relative positions of the components in their local reference system

• Differential coordinates should be based on the apparent places of both components, computed separately

 This will account for any geometric or light-propagation effects that shift the position of one component relative to the other 9

Δϕ"

B"

To"Earth"at"*me"t" sep"0" sep"1" i

A" sep 0 = (B−A) cos i sep 1 = (B−A) cos (i + Δϕ)

Suppose B−A = 100 arcsec, i=45°, and Δϕ = 20 arcsec = 0.00555°.

Then sep 0 − sep 1 = 70.7107 arcsec Plane"of"the" − 70.7038 arcsec = 6.9 mas sky"at"*me"t""""" Plane"of"the" (="*me"of" sky"at"*me"E" Position angles could also be affected, ephemeris"or" (="orbit’s"equinox)" depending on the relative orientations observa*on)" of the planes. 10 To"Earth,"assumed"at"infinity," at"*me"t"

B"

sep"2" sep"2" sep"1" i’ Earth"at"*me"t" d" δ" (distance"Earth"to"star"A)" A"

sep 1 = (B−A) cos i’ = cos (i + Δϕ) = 70.7038 arcsec (from previous figure)

sep 1 (B−A) cos i’ sep 2 = arctan = arctan d − δ d − (B−A) sin i ’ Plane"of"the"

sky"at"*me"t" Again suppose B−A = 100 arcsec, i=45°, i’=45°.0555°, (="*me"of" and d = 100 pc. ephemeris"or" observa*on)" Then sep 1 − sep 2 = 70.7038 arcsec − 70.7281 arcsec = −24.2 mas 11 aberrated geometric Aberration moves the apparent places of positions positions the two stars slightly differently on the sky, which can affect both separations B" B" v/c" and position angles. Differential projected"at"B" aberration can amount to sep"2" 1 mas for 10 arcsec separations and sep"3" 10 mas for 100 arcsec separations. In this case, with sep 2 = 70.2781 arcsec, A" A" v/c" sep 2 − sep 3 could be up to ≈ 7 mas. projected"at"A" B"

Earth’s"orbital"velocity" v"" A"

12 Computing Apparent Places of Single Stars

• α, δ, µα, µδ, π, vR for t0 p0 and v

• Apply space motion: p1(t) = p0 + v(t−t0)

• Correct for parallax: p2(t) = p1(t) − E(t)

• Compute gravitational light bending: p3(t) = f (M,p2,…)

• Apply aberration: p4(t) = p3(t) + E(t)/c

• Rotate, if necessary, to desired final reference system:

p5(t) = (t) p4(t) where (t) contains precession, nutation, etc.

• Decompose p5 into α, δ for t 13 Computing Apparent Places of Binary Stars

• α, δ, µα, µδ, π, vR for t0 p0 and v

• Apply space motion: p1(t) = p0 + v(t−t0)  Compute component offsets from 2-body code

• Correct for parallax: p2(t) = p1(t) − E(t)

• Compute gravitational light bending: p3(t) = f (M,p2,…)

• Apply aberration: p4(t) = p3(t) + E(t)/c

• Rotate, if necessary, to desired final reference system:

p5(t) = (t) p4(t) where (t) contains precession, nutation, etc. Separately for A & B & A Separatelyfor

14 • Decompose p5 into α, δ for t

How Different Are the Ephemerides?

Ephemeris results for 2488 orbits and 5 dates, compared to file computed independently using conventional approach:

• 703 (28%) have differences of 1 or more end-figures

 Over half of these are due to aberration

• 27 (1%) have difference of 2 or more end-figures

• 15 (0.6%) have differences of 6 or more end-figures

• 12 (0.5%) have differences of 10 or more end-figures

End figures are 1 mas in separation and 0.1 degree in position angle

15 ORB6 statistics

1600 1600 1400 1400 1200 1200 1000 1000 800 800 600 600 400 400 200 200 0 0 1.E-05 1.E-04 0.001 0.01 0.1 1 a (arcsec) π (arcsec)

1400 1200 1000 800 600 400 200 0

a (au) 16