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the main asteroid belt is a relic from the formation of the solar system . although much of its mass has been lost , it retains a great deal of information about solar system history and presents us with a laboratory in which we can study collisional processes that once operated throughout the circumsolar disk in which earth and the other planets were formed . one of the most straightforward observables constraining such processes is the asteroid belt s size - frequency distribution ( sfd ; bottke et al . the current main belt s sfd can be successfully modeled as the result of 4.5 billion years of collisional evolution @xcite . while such models fit the ` collisional wave ' set up by 100 km asteroids able to survive unshattered through the age of the solar system , they can not be observationally tested in the 100 meter size range . objects in this size range are very interesting , because they supply most near - earth asteroids and meteorites by shattering one another and/or migrating inward via yarkovsky and resonance effects @xcite . modern 8 - 10 meter telescopes can detect them , but monitoring them over many nights to determine an orbit requires a prohibitively large time investment for such powerful telescopes ( e.g. , 710 nights ; gladman et al . thus their distances and sizes remain unknown , and detailed analyses are confined to larger objects @xcite or use only rough statistical distances @xcite .    we present a method to obtain precise distances to main belt asteroids ( mbas ) using only two nights of observations . distances translate directly into absolute magnitudes and hence to sizes given a reasonable assumption for the albedo distribution . this method , which we refer to as rotational reflex velocity ( rrv ) , will greatly increase the efficiency of surveys aimed at probing collisional evolution in the solar system by measuring the sfds for extremely small mbas . we demonstrate rrv distance determination using a data set from the 0.9-meter wiyn telescope , which we have analyzed using digital tracking @xcite in order to enhance our sensitivity to faint asteroids . digital tracking is a method for detecting faint moving objects that was first applied to the kuiper belt ( e.g. bernstein et al . 2004 ) , and very recently has begun to be applied to asteroids @xcite . although the rrv distances we calculate herein are all based on our digital tracking analysis , the rrv method is equally useful for asteroids detected by more conventional means , or by other specialized methods such as those of @xcite and @xcite . suppose that at a given instant , an asteroid located a distance @xmath0 from an earth - based observer is moving with velocity @xmath1 , while the observer is moving with velocity @xmath2 ( e.g. , the orbital velocity of the earth ) . the angular velocity at which the observer sees the asteroid move relative to distant stars is given by :    @xmath3    where the @xmath4 subscript indicates the vector component perpendicular to the line of sight , so that @xmath5 is the projection of the asteroid s relative velocity onto the plane of the sky . although @xmath2 can be accurately calculated for any earth - based observation , the velocity @xmath1 of a newly discovered asteroid is always unknown initially , and hence the distance can not be calculated by simply plugging the measured value of @xmath6 into equation [ eq : bot ] . given appropriate measurements , however , we can isolate the component of @xmath6 that reflects the observer s motion around the geocenter due to earth s rotation , and from this calculate the distance . this is the essence of the rrv method for distance determination . the velocity @xmath2 of an observer on the surface of the earth can be expressed as the sum of earth s orbital velocity @xmath7 and the velocity @xmath8 with which the earth s axial rotation carries the observer around the geocenter . neglecting the slight asphericity of the earth , @xmath9 , where @xmath10 is the observer s terrestrial latitude and @xmath11 is the earth s equatorial rotation velocity of 1674.4 km / hr . for convenience , we define @xmath12 as the asteroid s velocity relative to the geocenter : @xmath13 . the angular velocity @xmath14 that would be measured by an ideal observer located at the geocenter then depends only on @xmath12 and the distance , but the angular velocity @xmath15 that is measured by a real observer based on the earth s surface depends also on @xmath8 . the two angular velocities are given by :    @xmath16    @xmath17    if we could measure @xmath14 , we could therefore calculate the distance :    @xmath18    where we have dropped the @xmath4 subscript , because it will henceforward apply to all physical velocities in our calculations . now suppose that the asteroid is observed near midnight on two different nights , that the two observations are separated by exactly one sidereal day , and that the position and angular velocity @xmath15 are recorded for each observation . the angular distance the asteroid moved between the two observations will thus be accurately known ; call this @xmath19 . because exactly one full rotation of the earth elapsed between the two observations , the observer s position relative to the geocenter is the same for both of them . thus , the average geocentric angular velocity of the asteroid in between the two measurements is @xmath20 , where @xmath21 is the elapsed time between the observations : one sidereal day . let the measured values of @xmath15 on the first and second nights be @xmath22 and @xmath23 , and similarly let the perpendicular rotational velocities ( which are obtained by calculation , not measurement ) be @xmath24 and @xmath25 . we can then evaluate the difference between geocentric and observer angular velocities twice : the average of @xmath26 and @xmath27 will be a factor of @xmath28 more precise than a single measurement if the uncertainty on @xmath14 measurement is much smaller than on @xmath22 and @xmath23 . this is likely to be the case , since @xmath14 is based on a longer temporal baseline . the distance is then given by :    @xmath29    so far we have assumed that @xmath12 and @xmath0 show no appreciable change over the 24 hour period of the measurements , so that @xmath26 and @xmath27 are effectively two measurements of the same quantity . we will now determine the errors that result when this assumption is violated . we will first consider changes in @xmath12 and @xmath0 that are linear in time : that is , when the first time derivatives @xmath30 and @xmath31 are significant but the second derivatives @xmath32 and @xmath33 are not . we will parameterize the change in @xmath12 by @xmath34 , and the change in @xmath0 by @xmath35 . for convenience , we will also use @xmath36 since the observations do not have to be taken exactly one sidereal day apart , and hence the earth s projected rotational velocity may vary . for well - optimized observations @xmath37 should always be close to zero . note that while @xmath38 and @xmath37 are unit - less , fractional changes , @xmath39 still has units of linear velocity . we then obtain :    @xmath40    we substitute these into equation [ eq : av01 ] . retaining terms only to second order in the variational quantities @xmath39 , @xmath38 , and @xmath37 , after a good deal of algebra we find that the fractional error on the calculated distance is :    @xmath41    for main belt asteroids @xmath38 is always very small ( i.e. @xmath42 ) . this renders @xmath37 innocuous even though it can reach 0.3 for poorly - planned observations ( section [ sec : nonopt ] ) . the velocity change @xmath39 approaches zero for asteroids at opposition , where the accuracy of equation [ eq : av01 ] can be better than @xmath43 , but even months from opposition the error remains only of order @xmath44 . equation [ eq : av01 ] is therefore very accurate under all conditions where the required input data could reasonably be obtained , and the uncertainties even of highly accurate measurements ( section [ sec : measprec ] ) will generally be larger than the equation s intrinsic error . we caution the reader against a much less accurate alternative to equation [ eq : av01 ] that may appear more attractive because it is obtained simply by applying equation [ eq : vel2 ] individually to each night and then averaging the resulting two distance measurements :    @xmath45    despite its intuitive simplicity , this equation should never be used . its fractional error expansion contains the term @xmath46 , which has very large values for objects more than a few days from opposition , as illustrated by figure [ fig : reflexerrors ] . by contrast , for equation [ eq : av01 ] the precursors of the @xmath46 term and several other error terms cancel in the denominator , leaving only the very much smaller error terms given by equation [ eq : err02 ] . equation [ eq : av00 ] should therefore be avoided , and equation [ eq : av01 ] ( or its generalized form , equation [ eq : newav01 ] ) should always be used for rrv distances . for simplicity , the discussion in section [ sec : math ] assumed the observer could make instantaneous angular velocity measurements . in practice , of course , angular velocity measurements must be made over a period of time . this has very little effect on our mathematical derivation . the only change is that the projected velocities of the observer relative to the geocenter , @xmath24 and @xmath25 , should not be considered as instantaneous but rather as averages over the same period of time in which the angular velocities @xmath22 and @xmath23 were measured .    positions and angular velocities of asteroids are measured in terms of right ascension ( ra ) and declination ( dec ) on the celestial sphere , and thus are two dimensional . for example , an observer could measure an asteroid as having an angular velocity of -37.36 arcsec / hr in ra ( the negative sign indicating westward rather than eastward motion ) and 10.32 arcsec / hr in dec , with the positive sign indicating the object is moving toward celestial north . in principle , independent versions of equation [ eq : av01 ] could be constructed for each dimension , with angular velocities in ra corresponding exactly to projected east - west motion of the observer and dec velocities corresponding to projected north - south motion . however , the true rotational velocity of the observer is always strictly east - west , and although its projection @xmath8 can have a north - south component , this component is usually small  i.e. , less than 10% of the east - west component for observations with mean hour angle in the range @xmath47 targeting asteroids within @xmath48 of the celestial equator . as we will find below , for our test observations only the ra velocity components are large enough to supply useful distances . we expect this will generally be the case .    in most asteroid surveys , many fields are observed , and each field is visited at least three times per night . each visit yields a single celestial position of each detected asteroid . two observations of each asteroid on each night will suffice for our purposes . using the traditional symbols @xmath49 and @xmath37 for ra and dec , suppose that on night 1 we obtain observations @xmath50 at time @xmath51 and @xmath52 at time @xmath53 , and similarly on night 2 . then we can also define mean positions and the mean time @xmath54 , @xmath55 , and @xmath56 , with the analogous quantities being calculated for night 2 . note that the times here are measured on a continuous sequence and not reset when the date changes . for simplicity , at present we will suppose further that the time of each measurement on night 2 is exactly one sidereal day later than the time of the corresponding measurement on night 1 : thus @xmath57 one sidereal day . the required angular velocities are then given by the following equations , where for simplicity we show only the ra component :    @xmath58    @xmath59    @xmath60    where the denominator of the last equation is of course also equal to one sidereal day . n , roughly the latitude of mauna kea ) , but they are separated by an elapsed time of 3.989 hr , corresponding to 60@xmath61 of sidereal rotation . the rotational velocity of earth s surface at 20@xmath61 n lat is 1573.4 km / hr . the effect on the asteroid s measured positions is determined by the projected distance moved between the observations , which has eastward and northward components as labeled . the projected rotational velocity @xmath8 that should be used in the rrv distance calculation ( equation [ eq : av01 ] or equation [ eq : newav01 ] ) is therefore @xmath62 km / hr eastward and @xmath63 km / hr northward . [ fig : earth ] ]    the only other quantities needed for the distance calculation are @xmath24 and @xmath25 , the observer s projected rotational velocities relative to the geocenter . as illustrated by figure [ fig : earth ] , the observer s projected motion relative to the geocenter traces out a segment of an ellipse whose axis ratio is the sine of the asteroid s declination . the projected distance of the observer from the geocenter is given by :    @xmath64    where @xmath65 and @xmath66 are positive toward the east and north ; @xmath67 is the observer s latitude on the earth ; @xmath37 is the declination of the asteroid ; @xmath68 is the time of the observation ; @xmath69 is the time when the asteroid stands at zero hour angle for the observer ; and @xmath70 is the length of a sidereal day . note that the argument @xmath71 is the hour angle at which the observer sees the asteroid : at @xmath72 the asteroid crosses the meridian and stands closest to the observer s zenith and in the best sky position for observing .    for observations at @xmath51 and @xmath53 , the projected linear velocity @xmath24 in the east - west ( @xmath65 ) and north - south ( @xmath66 ) directions is :    @xmath73    @xmath74    figure [ fig : earth ] illustrates a specific example of two observations of an asteroid at -12@xmath61 dec , made from a single observing site at 20@xmath61 n lat on the earth s surface . the first observation is made when the asteroid is rising , and the second about four hours later , shortly after it has crossed the meridian . the rotational velocity of earth at 20@xmath61 n lat is 1573.4 km / hr and is of course strictly eastward , but projecting the velocity into a plane perpendicular to the line of sight reduces the absolute magnitude and introduces a north - south component ( except for asteroids exactly at 0@xmath61 dec ) . in this example the projected velocities east and north are 1410.3 km / hr and 106.2 km / hr . since celestial ra and dec correspond exactly to terrestrial longitude and latitude , the east - west physical velocity of equation [ eq : proj2x ] can be combined with the ra components of the angular velocities from equations [ eq : meas01][eq : meas03 ] and plugged directly into equation [ eq : av01 ] to yield the distance . we will now provide a simplified but basically realistic example . suppose the mba illustrated by figure [ fig : earth ] is being observed at opposition . it will be in its retrograde loop , and hence its motion in ra will be negative ( westward ) . the angular velocity @xmath75 measured by the observer will be faster to the west than the geocentric angular velocity @xmath14 , because on the night side of the earth the rotational velocity adds to the orbital velocity , making the asteroid fall back westward at a faster rate . arcsec / hr and @xmath77 arcsec / hr . the rotational reflex velocity , given by the difference @xmath78 , is then 1 arcsec / hr or @xmath79 rad / hr . taking the physical and angular velocities to be exactly the same on the second night for simplicity , equation [ eq : av01 ] reduces to :    @xmath80    with equation [ eq : calc01 ] we have calculated the distance to an asteroid based on the known properties of earth s rotation . interestingly , @xcite have developed a closely analogous method for determining the distance to a kuiper belt object using the known properties of earth s _ orbit_. just as our derivation does not ( explicitly ) include the effects of earth s orbit , that of @xcite does not include the effects of earth s rotation . neglecting earth s orbit works in our case because , as we have demonstrated in section [ sec : math ] , the dominant errors due to orbital acceleration of both the earth and the asteroid cancel when the correct form of the distance equation is used . similarly , @xcite can safely neglect earth s rotation because the kuiper belt objects whose distances they aim to measure are sufficiently far away that their rrv signature is negligible . it follows that the method of @xcite should not be directly applied to measuring asteroid distances without some form of correction for the rrv signal , although ( as they point out ) it is useful for distinguishing true kuiper belt objects from asteroids that have similar angular velocities for a few days near their turnaround points .      up to now we have assumed for simplicity that the observations obtained on successive nights are exactly one sidereal day apart . while it is not difficult to plan and execute observations that satisfy this criterion to within a few minutes , clouds or instrument failures could intervene , or one could be processing an archival data set that was not taken with rrv distances in mind . in such cases the mean times @xmath81 and @xmath82 may be separated by up to a couple of hours less or more than one sidereal day . a direct calculation of the geocentric angular velocity @xmath14 then becomes impossible . in its place , we must use instead the mean angular velocity across the two nights , defined by :    @xmath83    we will define three additional quantities . the time @xmath84 is the moment exactly one sidereal day before @xmath82 ( in the ideal case , @xmath81 would have been equal to @xmath84 ) . the projected rotational velocity @xmath85 is the observer s mean projected rotational velocity over the time interval between @xmath81 and @xmath84 . finally , the fractional parameter @xmath86 is given by :    @xmath87    note that @xmath86 goes to zero in the case of optimally - timed measurements . the distance formula analogous to equation [ eq : av01 ] becomes :    @xmath88    the error terms for equation [ eq : newav01 ] are the same as those given in equation [ eq : err02 ] , with the addition of an @xmath89 term . for main belt asteroids , this will always be negligible . in closing our discussion of error in the case of non - optimal observation timing , we note that @xmath37 , the fractional change in projected rotational velocity from night 1 to night 2 , is likely to be considerably larger in this case , up to 0.3 . however , in the error terms for equation [ eq : newav01 ] it is always multiplied by @xmath38 , which is always of order @xmath44 or smaller for mbas , and thus should not constitute a major source of error . equation [ eq : newav01 ] is therefore the central formula for asteroid distance determination with rrv . in addition to our analytical error calculation ( equation [ eq : err02 ] ) , we have probed the errors of our approximations numerically using ephemerides of known objects from the jpl horizons ephemeris generator . note that no actual observations are involved in this test , only ephemeris positions and velocities for objects with well - known orbits . in figure [ fig : reflexerrors ] we plot the error of equation [ eq : newav01 ] as determined by this test . in contrast to our analytical error calculations , this test intrinsically includes the contributions of second and higher - order time derivatives of @xmath0 and @xmath12 , since it uses jpl horizons ephemerides for real objects . the plotted errors are affected by roundoff error in the ephemerides ( accurate only to 0.15 arcsec in ra ) , which introduces pseudo - noise such that only the lower envelope of the plotted points can be meaningfully compared with our analytical error estimates . nevertheless , the plots show that equation [ eq : newav01 ] produces accurate distances out to at least 60 days from opposition ; that accurate distances can be obtained from observations that are not optimally timed ; and that the second - order time derivatives @xmath32 and @xmath33 do not introduce significant error . figure [ fig : reflexerrors ] , although based on calculated ephemerides and not actual data , shows that relatively small position errors can create distance inaccuracies at the level of a few percent . we now consider the measurement precision that is necessary to yield distances to various levels of accuracy .    in equation [ eq : calc01 ] , the difference between the night - to - night mean angular velocity in ra ( @xmath90 ) and the angular velocity in ra observed within a given night ( @xmath91 ) was one arcsec / hr , and this rotational reflex velocity corresponded to a distance of about 2 au . this is typical for observations obtained near opposition at terrestrial latitudes of 2030@xmath61 . the fractional uncertainty of the distance is equal to that of @xmath92 . this will generally be dominated by the uncertainty in @xmath91 , since it is calculated within a single night and is thus based on a smaller temporal baseline than @xmath90 . thus , if @xmath91 is measured with an accuracy of 0.1 arcsec / hr , we should expect the distance to have an accuracy of only 10% at 2 au . at 1 au , the rrv signature is twice as large : @xmath92 will therefore be about 2 arcsec / hr , and the calculation will be accurate to 5% .    as we will now illustrate , one can usually obtain angular velocity measurements with better than 0.1 arcsec / hr uncertainties in practice . in our current paradigm , @xmath91 is based on two measurements of celestial positions separated by a time @xmath93 ( i.e. , equation [ eq : meas01 ] or [ eq : meas02 ] ) . if the uncertainty on ra for these measurements is @xmath94 , the uncertainty on @xmath91 will be :    @xmath95    based on our experience , well - sampled ( e.g. @xmath96 arcsec / pixel ) images taken in @xmath97 arcsecond seeing can yield @xmath94 = 0.030.05 arcsec for bright objects ( e.g. , those at least several times brighter than the 10@xmath98 detection threshold ) . for faint objects near the detection limit , uncertainties of @xmath99 arcsec are more typical ) rather than being obtained from two positions . ] a temporal baseline of 5 hr is easy to obtain near opposition . thus , we can expect @xmath100 = 0.0080.014 arcsec / hr for bright objects and @xmath100 = 0.03 arcsec / hr for fainter objects . at 2 au , distances will therefore be accurate to about 3% for faint objects and 11.5% for bright objects , while at a distance of 1 au , the accuracy should be about 1.5% even for faint objects . these accuracies are sufficient for good analyses of the size statistics of small mbas , especially given that their albedos are unknown and must be treated as a statistical distribution spanning a factor of @xmath10110 @xcite . where the objects are faint or distant ( e.g. 35 au ) , or where extremely accurate distances are desired , it may be possible to obtain smaller values of @xmath94 by obtaining each ra measurement not from a single image but from the average over a set of images acquired close together in time . extending the same principle further , we can use the technique of digital tracking to obtain accurate measurements for objects too faint even to be detected in individual images , as we describe below . the observations we use to demonstrate the usefulness of rotational reflex distances for asteroids were obtained using digital tracking , as described in our companion paper @xcite . these data consist of 126 two - minute exposures of a single starfield that we acquired using the wiyn 0.9-meter telescope at kitt peak on the night of april 19 , and 130 identically acquired images from the following night . very briefly , a digital tracking search involves shifting and stacking such sets of images to reveal moving objects too faint to be detected on any individual frame . a separate trial stack is produced for each angular velocity vector in a finely - sampled grid that spans the full range of possible sky velocities for the target population . for mbas , a digital tracking data set can span only one night @xcite , hence we analyze our april 19 and april 20 observations independently . digital tracking has been used to great effect for kuiper belt objects ( e.g. allen et al . 2001 , fraser & kavelaars 2009 , and many others ) , but has not typically been used for faster - moving asteroids where a larger numbers of trial vectors must be probed . it is now computationally tractable even for asteroids , however , and produces a factor of @xmath102 increase in sensitivity over conventional methods . this sensitivity increase enables us to detect 215 asteroids within the 1-degree field of our test observations , despite using only a small , 0.9-meter telescope . we obtained precise angular velocity measurements on both nights for 197 of these asteroids , including 48 previously known objects with accurate orbits . these last allow us to test the accuracy of rrv distance measurements using real data . the geocentric angular velocity @xmath14 of an asteroid detected on two subsequent nights is determined in exactly the same way with digital tracking observations as with conventional data . the determination of the observed angular velocities @xmath15 on each individual night is different , however  particularly because digital tracking enables the accurate measurement of asteroids that can not even be detected on an individual image . any asteroid detection in a digital tracking search occurs on a particular trial image stack , which corresponds to a particular angular velocity . thus , for example , our automated digital tracking routine might search for objects with angular velocities between -50 and -20 arcsec / hr in ra and -10 and + 20 arcsec / hr in dec , using a grid spacing of 0.2 arcsec / hr , and might detect a particular asteroid as a bright point source on a trial stack whose shifts correspond to an angular velocity of -41.2 arcsec / hr in ra and 11.8 arcsec / hr in dec . we improve on the relatively crude angular velocity measurement that is implicit in such a detection by probing a new , much more finely spaced grid of angular velocity vectors , using small postage - stamp images centered on the detected asteroid to make the search computationally tractable . for each trial stack in this finely sampled grid , we calculate the measured flux of the asteroid within a small aperture of radius roughly equal to the half - width of the point spread function ( psf ) of our image . we then perform a 2-d quadratic fit to the measured flux as a function of the trial angular velocity . the angular velocity at which the quadratic fit reaches its peak value then constitutes an accurate measurement of @xmath15 for that asteroid , and the uncertainty on @xmath15 is derived from the uncertainty of the quadratic fit , which is naturally larger for faint objects that are more affected by sky background noise .    all the inputs required to calculate the distance using equation [ eq : newav01 ] are now available except @xmath8 . as in the case of discrete observations , equation [ eq : proj1 ] indicates how @xmath8 should be calculated , but there is one subtle aspect . the appropriate value of @xmath8 is not a projected distance divided by an elapsed time as it was in the discrete case ( figure [ fig : earth ] ) , but neither is it the time average of the projected velocities @xmath103 and @xmath104 . instead , the value of @xmath8 that corresponds to the angular velocity measured by digital tracking is given by the slopes of the best linear fits to @xmath65 and @xmath66 as functions of time . figure [ fig : proj1 ] illustrates the difference between linear fit velocities and average velocities .    for our data , the linear fit velocities of the kitt peak observer were 1340.0 km / hr eastward and -4.8 km / hr northward on april 19 ; and 1337.3 km / hr eastward and -16.3 km / hr northward on april 20 ; by contrast the average velocities were 1286.6 km / hr eastward and -6.9 km / hr northward on april 19 and 1283.0 km / hr eastward and -14.6 km / hr northward on april 20 . in this case , the north - south velocities are negligible for practical purposes . our observations on 2013 april 19 span a temporal range of 5.72 hr . those obtained on april 20 have the same range , and are centered 23.968 hr later in time . as this interval is only 2 minutes longer than a sidereal day , the timing of our observations is almost exactly optimal as defined in section [ sec : opt ] : the parameter @xmath86 from equation [ eq : meas08 ] is only 0.0014 . we detected a total of 215 asteroids through digital tracking analysis of our observations in a field of view only slightly larger than 1 square degree . we reported positions and brightnesses for all of these objects to the minor planet center . of these 215 mbas , 202 were detected on both april 19 and april 20 , and 197 of these two - night objects had sufficiently accurate measurements on both nights for good distance calculations . of these , 48 were previously discovered objects with accurately known orbits , nine were previously discovered objects with poorly known orbits prior to our observations , and the remaining 140 were new discoveries for which we received designations from the minor planet center . @xcite provides further details . we have used equation [ eq : newav01 ] to calculate distances for all 197 objects with sufficient measurements . the 48 objects in this sample with well - known orbits allow us to test the accuracy of rotational reflex distances using real data . the mean absolute error of our calculated distances for these objects is 0.042 au , and the mean absolute fractional error is 2.2% . as expected , errors are smaller for more nearby objects . distances with 2% accuracy are more than sufficient to analyze the size statistics of mbas . nevertheless , even better results are possible , as we will see below . close investigation reveals some evidence of systematic error in the distance calculations described in section [ sec : initdist ] . the weighted average signed fractional error for the 48 known objects is @xmath105 : i.e. , the calculated distances are 1.8% too small on average , and the offset is 10@xmath98 significant . it might be suspected that this bias is due to using the wrong value for @xmath8  i.e. , that we were wrong to use linear fit velocities rather than time - averaged velocities ( section [ sec : angdigi ] ) . this is easily demonstrated not to be the case , however . substituting averaged values in place of linear fit values for @xmath8 changes the result too much and in the wrong direction : the mean systematic offset goes from -1.8% to -5.2% . the true cause of the bias is more subtle , and points to a further interesting measurement that can be extracted from our data .    with digital tracking , our measurement of the single - night observed angular velocity @xmath15 is made over a period of time equal to the temporal span of our observations on each night . the instantaneous angular velocity over this interval is continuously changing , with the rotational reflex velocity itself being the dominant source of change . for example , the rotation of the earth causes the westward sky motions of the asteroids to be slightly faster near midnight than they are at the beginning and end of each night s observations ( see figure [ fig : proj1 ] ) . as we stated in section [ sec : angdigi ] and illustrated by figure [ fig : proj1 ] , we can account for the acceleration and curvature of an asteroid s observed motion by using a value for @xmath8 that is obtained by a linear fit to the observer s projected coordinates @xmath106 and @xmath107 . this assumes , however , that the angular velocity measurement is effectively an _ unweighted _ fit to angular position as a function of time . such is not necessarily the case . for example , the images taken near midnight , when the westward sky motion is at its fastest , may be more sensitive due to better seeing or lower atmospheric extinction . the digital tracking stack will then effectively weight them more highly and produce a mean angular velocity that is systematically too fast . this effect would produce a systematic underestimation of the asteroids geocentric distances : exactly what we observe for known asteroids in our data .    the effective weighting of different images will not necessarily be the same for different asteroids . an asteroid that passes too close to a bright star near midnight will only be measured near the beginning and end of the night , resulting in a measured angular velocity slower than the true average value . asteroids can also exhibit significant brightness changes during the night due to their own rotation , which will change the effective weighting of different images in the final measurement of their angular velocities . if asteroids actually moved in straight lines at constant velocity , none of these weighting effects would introduce error into our angular velocity determinations . if we could _ predict _ the curvature and acceleration of each asteroid s track due to the earth s rotation , and apply small shifts to each image in our digital tracking stacks in order to linearize the motion , we would therefore remove the bias in our angular velocity determinations . we now describe how to do this . the angular distances between the position an observer measures for an asteroid and the position it would have if measured from the geocenter are simply a reflection of equation [ eq : proj1 ] , scaled by the distance :    @xmath108    averaged over the span of a digital tracking integration , @xmath109 and @xmath110 contribute a constant offset to the measured position of an asteroid , and their first time derivatives @xmath111 and @xmath112 produce a constant offset ( i.e. the rrv signal itself ) to its measured angular velocity . neither of these constant offsets concerns us here : we wish to isolate and remove only the component that represents curvature and acceleration . thus , we create new functions @xmath113 and @xmath114 that behave exactly like @xmath115 and @xmath116 except that over the span of our digital tracking integrations , their means and the means of their first time derivatives are zero :    @xmath117    where the notation @xmath118 denotes the time - average of the enclosed quantity over the span of a digital tracking integration . the left panel of figure [ fig : curve ] presents equation [ eq : curve2 ] in graphical form , plotting the curved track of an asteroid in @xmath119 space over the course of our april 19 observations . the calculation of @xmath120 and @xmath121 of course requires knowledge of the asteroid s distance @xmath0 , but the 2% accurate values from section [ sec : initdist ] are more than sufficient for this purpose . thus , we adjust the methodology described in section [ sec : angdigi ] for measuring precise angular velocities from digital tracking data : we apply additional shifts equal to @xmath122 to each postage stamp image before the final stack . this straightens the asteroid s track so that curvature can no longer influence the calculated angular velocity . the distance can be recalculated based on the resulting new , de - biased angular velocities . note that because equation [ eq : curve2 ] subtracts _ average _ reflex velocities , we must now use averaged rather than linear - fit values for @xmath8 ( section [ sec : angdigi ] and figure [ fig : proj1 ] illustrate how the velocities differ ) . compared to the calculation in section [ sec : initdist ] , our curvature correction reduces the mean absolute error of the distances for known objects from 0.042 au to 0.032 au , and the mean absolute fractional error from 2.2% to 1.6% . more significantly , the new calculation changes the weighted average signed fractional error from @xmath105 to @xmath123 . the systematic offset is thus reduced by a factor of six , and the new value of -0.3% is statistically consistent with zero . the left panel of figure [ fig : distmag ] illustrates the precision of our measured distances for known objects . we emphasize that these distance measurements were entirely calculated from first principles and known quantities describing the earth . we have used the known asteroids in our data to quantify our errors , but not to determine any of the parameters used in the distance calculation .      beyond removing systematic errors in our measurements of asteroids mean angular velocities , we can in principle use equation [ eq : curve2 ] to _ measure _ the curvatures of asteroid tracks within just a single night s data . this is an extremely challenging measurement because the curvatures are so small : the rms curvature amplitude , defined as the root mean of @xmath124 , is less than 0.1 arcsec for an asteroid at 2 au . we attempt to measure it using our curvature - corrected postage - stamp analysis as described in section [ sec : two_precise ] , but instead of calculating @xmath125 and @xmath126 based on a previously - calculated approximate distance , we probe a range of rms curvature amplitudes for each asteroid . for each trial curvature amplitude we calculate @xmath125 and @xmath126 , apply the appropriate shifts to the postage stamp images , and perform a quadratic fit to flux as a function of angular velocity as described in section [ sec : angdigi ] . we compare the peak asteroid flux values produced by the quadratic fits for the different trial curvature amplitudes , and identify the curvature amplitude that yields the highest peak asteroid flux . this should be the curvature value that resulted in all of the individual asteroid images being most accurately registered . as the curvature amplitudes map uniquely to distances ( equations [ eq : curve1 ] and [ eq : curve2 ] ) , our identification of the best - fit amplitude constitutes a distance measurement , in principle . however , the uncertainty of curvature amplitudes measured from the current data set is large enough that we do not choose to calculate individual curvature - based distances . instead , we plot the best - fit curvature amplitudes as a function of the known rrv distances calculated in section [ sec : two_precise ] . a clear inverse relationship emerges , as predicted by equation [ eq : curve1 ] . the relationship gets tighter when we reject the minority of asteroids with out - of - range best - fit curvature values and all asteroids fainter than @xmath127 mag , which can not be measured as accurately . the 48 objects meeting these criteria in our april 19 observations are plotted in the right - hand panel of figure [ fig : curve ] , which shows the rms curvature amplitude as a function of geocentric distance in au , and compares the data with the predicted curve from equations [ eq : curve1 ] and [ eq : curve2 ] . although there is a mild systematic offset , the slope of the best linear fit to curvature as a function of @xmath128 for these 48 asteroids is 9@xmath98 significant , matches the predicted value to within 0.4@xmath98 , and has an intercept only 1.4@xmath98 away from the predicted value of zero . the rms error of the measured curvature amplitudes relative to their predicted values is only 0.03 arcsec . thus we have effectively performed astrometry of moving , 20th magnitude objects with a precision of 30 milli - arcsec using only an 0.9 m telescope .    in principle , our success at measuring asteroid curvatures within a single night s data means that digital tracking can be used to measure the geocentric distances to asteroids based on observations from only one night . however , our current curvature measurements have uncertainties that are too large to yield useful distances for individual objects : we have been able to demonstrate the efficacy of our curvature - measuring methodology only by recourse to an ensemble of 48 asteroids with distances known by other means . with a larger telescope , more accurate curvature measurements of fainter asteroids would be possible , especially in good seeing . these could conceivably yield meaningful distance measurements for unknown main - belt asteroids over just a single night , albeit only for objects considerably brighter than the digital tracking detection limit . by contrast , the two - night rrv calculations of sections [ sec : initdist ]  [ sec : two_precise ] yield precise distances even for faint objects just above the detection threshold . there is therefore little reason to use the curvature method for asteroids measured on more than one night . however , where only one night s data is available , distances based on digital - tracking curvature measurements could be very valuable . this is particularly true in at least two cases that may arise in future surveys . the first is the case of an neo survey where some objects  perhaps due to their fast motion  have inadvertently been measured on only one night . being much closer to the earth than mbas , neos will show much larger and more easily measurable curvature , which is likely to yield useful distances . such distances could enable the inclusion in a statistical analysis of objects that were otherwise unusable due to their one - night status . the second case is the one - night detection of a nearby neo moving at nearly the same space velocity as earth , such that its slow angular velocity mimics that of a much more distant object . this scenario is statistically rare but troubling , because such an object would likely be overlooked in an neo survey and yet could be an incoming earth - impactor . if the discovery survey used digital tracking , a curvature analysis would reveal the object s actual , very small geocentric distance . we note in closing that curvature measurements can in principle be performed on asteroids detected with conventional methods rather than digital tracking , but they will normally be less accurate because of the smaller number of images available .      while our current data only suggest the possibility of curvature - based distance measurements from a single night , they allow precise two - night distance measurements from rotational reflex velocity ( section [ sec : two_precise ] ) . these precise distance measurements enable us to calculate the absolute magnitudes of each of our detected asteroids , and hence their physical diameters modulo the uncertainty in albedo . the histograms of these values are shown in the right - hand panel of figure [ fig : distmag ] . at least half of the asteroids we have measured are smaller than 1 km . by contrast , only a small fraction of known mbas are in this size range . the smallest objects we have detected are 300 meters in diameter under the assumption of low , 5% albedo , but could be as small as 150 meters if their albedo is 25% . the statistics of mbas in this size range are already known @xcite based on a sample several times larger than we present herein . given this , the detailed completeness analysis that would be required to convert our detections into a measurement of the sfd is not worthwhile . the aims of the current work and our companion paper @xcite are instead the validation of the rrv technique and of digital tracking for asteroids , respectively . similar observations using a larger telescope , however , would extend to much smaller objects and accurately measure the sfd of a previously unexplored size regime in the main belt . we have described how the reflex angular velocity of asteroids due to earth s rotation can be used to determine the distances to main belt asteroids based on only two nights of observations . we refer to this as the rotational reflex velocity ( rrv ) method for measuring asteroid distances ( equation [ eq : newav01 ] ) . the required approximations are accurate to about @xmath44 ( equation [ eq : err02 ] ) , and measurement uncertainties are typically 1 - 3% . such distances can be used to calculate precise size statistics of small main belt asteroids using a much smaller investment of time on a large telescope than has previously been required . accurate rrv distances may be calculated either from conventional asteroid - search observations , from data analyzed by the technique of digital tracking @xcite , or from any other specialized asteroid - observation technique ( e.g. milani et al . 1996 or gural et al . 2005 ) that accurately measures the celestial coordinates of the asteroids . we have tested rrv distance determination with a data set acquired on april 19 and 20 , 2013 using the wiyn 0.9-meter telescope at kitt peak . using measurements based on digital tracking , we have calculated distances to 197 asteroids in this data set . while the majority of these were new discoveries in our data , 48 of them are previously known objects with accurate orbits . these allowed us to test the accuracy of rotational reflex distances . a preliminary analysis yielded distances with a mean fractional error of only 2.2% . without digital tracking , our detections would be confined to substantially brighter asteroids ( or , the observations would require a larger telescope ) , but the accuracy of rrv distances for detected objects would likely be at least this good ( see section [ sec : measprec ] ) . we have identified a 1.8% systematic error in our preliminary rrv distances . we have linked this error to the curvature of the asteroids observed tracks that results from earth s rotation . this curvature , combined with the non - uniform sensitivity of images obtained under conditions of different seeing and airmass , causes a slight bias in the angular velocity measurements obtained from our digital tracking image - stacks . finding our initial distance measurements more than sufficient to predict curvature amplitudes , we have calculated the form and amplitude of the curvature for each of our asteroids . we have applied a curvature correction to our image stacks to linearize the asteroid motions . re - calculating the distances using angular velocities from curvature - corrected image stacks reduces the systematic bias from 1.8% to @xmath129% , and thus eliminates it as a significant effect . the corrected distances have a mean fractional error of only 1.6% for the 48 known objects in our data . note that such a curvature correction would not be needed for rrv distances based on asteroids detected on discrete images . the calculation of accurate rrv distances would therefore be simpler for conventional asteroid detections , although digital tracking can detect much fainter objects .    besides correcting our digital tracking angular velocity measurements for the curvature due to earth s rotation , we have attempted to measure this curvature by creating several different image stacks for each asteroid at a range of different curvature amplitudes . as the curvature amplitude is proportional to @xmath130 , this constitutes a rotational reflex velocity measurement of an asteroid s distance based on just a single night s data . we find that we can indeed measure the curvature amplitudes for asteroids brighter than @xmath131 mag with a precision of about 30 milli - arcsec . the measured curvature values for 48 asteroids brighter than this limit show the expected @xmath130 dependence at 9@xmath98 significance , and the best - fit slope is within 0.4@xmath98 of the predicted value . while distance measurements based on single - night curvature amplitudes are too noisy to be useful for asteroids in our current data set ( and are not needed given our highly precise two - night values ) , single - night distances could be valuable in specific cases for future surveys . in particular , curvature measurements can identify nearby objects moving at slow angular velocities characteristic of a much more distant population . our precise distances from two - night rrv measurements allow us to calculate absolute magnitudes and hence approximate diameters for our newly discovered asteroids . our faintest objects have @xmath132 mag and hence diameters of 130300 meters depending on their albedo . while the current census of the main belt becomes substantially incomplete at a diameter of about 2 km , we have detected dozens of new asteroids in the 200500 meter size range with an 0.9 m telescope . based on observations at kitt peak national observatory , national optical astronomy observatory ( noao prop . i d : 2013a-0501 ; pi : aren heinze ) , which is operated by the association of universities for research in astronomy ( aura ) under a cooperative agreement with the national science foundation . this publication makes use of the simbad online database , operated at cds , strasbourg , france , and the vizier online database ( see @xcite ) . we have also made extensive use of information and code from @xcite . we have used digitized images from the palomar sky survey ( available from http://stdatu.stsci.edu/cgi-bin/dss_form ) , which were produced at the space telescope science institute under u.s . government grant nag w-2166 . the images of these surveys are based on photographic data obtained using the oschin schmidt telescope on palomar mountain and the uk schmidt telescope . allen , r. l. , bernstein , g. m. , & malhotra , r. 2001 , , 549 , l241 bernstein , g. & kushalani , b. 2000 , , 120 , 3323 bernstein , g. m. , trilling , d. e. , allen , r. l. , brown , m. e. , holman , m. , & malhotra , r. 2004 , , 128 , 1364 bottke , w. f. jr . , durda , d. d. , nesvorn , d. , jedicke , r. , morbidelli , a. , vokrouhlick , d. , & levison , h. 2005 , icarus , 175 , 111 bottke , w. f. jr . , durda , d. d. , nesvorn , d. , jedicke , r. , morbidelli , a. , vokrouhlick , d. , & levison , h. f. 2005 , icarus , 179 , 64 de ela , g. c. & brunini , a. 2007 , , 466 , 1159 farinella , p. , froeschl , chr . , froeschl , cl . , gonczi , r. , hahn , g. , morbidelli , a. , & valsecchi , g. b. 1994 , nature , 371 , 314 fraser , w. c. & kavelaars , j. j. 2009 , , 137 , 72 gladman , b. j. , davis , d. r. , neese , c. , jedicke , r. , williams , g. , kavelaars , j. j. , petit , j - m . , scholl , h. , holman , m. , warrington , b. , esquerdo , g. , & tricarico , p. 2009 , icarus , 202 , 104 gural , p. s. , larsen , j. a. , & gleason , a. e. 2005 , , 130 , 1951 heinze , a. n. , metchev , s. , & trollo , j. 2015 , , submitted masiero , j. r. , mainzer , a. k. , bauer , j. m. , grav , t. , nugent , c. r. , & stevenson , r. 2013 , , 770 , 7 milani , a. , villani , a. , & stiavelli , m. 1996 , earth , moon , and planets , 72 , 257 ochsenbein , f. , bauer , p. & marcout , j. 2000 , , 143 , 23o pravec , p. , harris , a. w. , kusnirk , p. , galad , a. , & hornoch , k. 2012 , icarus , 221 , 365 press , w. h. , teukolsky , s.a . , vetterling , w. t. , & flannery , b. p. 1992 , numerical recipes in c ( second edition ; new york , ny : cambridge university press ) yoshida , f. , nakamura , t. , watanabe , j. 2003 , pasj , 55 , 701 yoshida , f. & nakamura , t. 2007 , p&ss , 55 , 1113 zhai , c. , shao , m. , nemati , b. , werne , t. , zhou , h. , turyshev , s. g. , sandhu , j. , hallinan , g. , & harding , l. k. 2014 , , 792 , 60
we present a method for calculating precise distances to asteroids using only two nights of data from a single location  far too little for an orbit  by exploiting the angular reflex motion of the asteroids due to earth s axial rotation . 
 we refer to this as the rotational reflex velocity method . 
 while the concept is simple and well - known , it has not been previously exploited for surveys of main - belt asteroids . 
 we offer a mathematical development , estimates of the errors of the approximation , and a demonstration using a sample of 197 asteroids observed for two nights with a small , 0.9-meter telescope . 
 this demonstration used digital tracking to enhance detection sensitivity for faint asteroids , but our distance determination works with any detection method . 
 forty - eight asteroids in our sample had known orbits prior to our observations , and for these we demonstrate a mean fractional error of only 1.6% between the distances we calculate and those given in ephemerides from the minor planet center . 
 in contrast to our two - night results , distance determination by fitting approximate orbits requires observations spanning 710 nights . 
 once an asteroid s distance is known , its absolute magnitude and size ( given a statistically - estimated albedo ) may immediately be calculated . 
 our method will therefore greatly enhance the efficiency with which 4-meter and larger telescopes can probe the size distribution of small ( e.g. 100 meter ) main belt asteroids . 
 this distribution remains poorly known , yet encodes information about the collisional evolution of the asteroid belt  and hence the history of the solar system .