Celestial Navigation – Correcting the FolklorePeter F. Swaszek

University of Rhode IslandKingston, RI USA

Richard J. HartnettU.S. Coast Guard Academy

New London, CT USA

Kelly C. SealsU.S. Coast Guard Academy

New London, CT USA

Abstract—Celestial navigation has been guiding the marinerfor hundreds of years; more recently, the GPS (GNSS, in general)has made 3-D positioning and precise time a commodity foreveryday use. With an increased awareness of the vulnerabilitiesassociated with any satellite-based system, there is a renewedinterest in non-satellite PNT systems. We have been reexaminingcelestial navigation from the vantage point of those conversant inGNSS language. Two aspects of these parallel approaches haveattracted our attention: how to choose which stars to process(similar to the GNSS satellite selection problem) and if (andat what accuracy) precise time can be estimated from celestialmeasurements.

This paper clarifies two common misconceptions about celestialnavigation: (1) that the stars selected for celestial navigation donot need to be “evenly” distributed across 360 degrees of azimuth,that multiple stars from nearly similar directions (i.e. Polaris andstars in the Big Dipper) can be very useful together in forming agood celestial fix; and (2) that while “star clocks” are advertised,their result is not precise time, and that celestial techniques canonly provide a low quality source of time information.

Index Terms—celestial navigation, DOP

I. INTRODUCTION

Celestial navigation, the art of determining one’s locationon the surface of the earth based upon measurements of acollection of stars, has been guiding the mariner for hundredsof years. Since 1802, Bowditch’s classic text “The NewAmerican Practical Navigator” (and its multiple revisions)has provided details of the methods of celestial navigationincluding the use of the sextant and examples of the sightreduction process [1]. Since the 1950’s electronic navigationsystems have overtaken celestial navigation, providing accu-rate measurements of position to the user. Early terrestrialsystems, such as Omega and Loran, provided 2-D positioningto maritime users; more recently, the GPS (and GNSS, ingeneral) has made 3-D positioning and time a commodity foreveryday use. Given increased awareness of the vulnerabilitiesassociated with any satellite-based system, there is currentlya renewed interest in non-satellite PNT (position, navigation,and time) systems.

In the world of GNSS, measurements are taken for eachsatellite and determining the user’s position (and time) in-volves solving the set of pseudorange equations; when overde-termined, this is in a least squares sense. While these equationsare nonlinear, they can be solved via an iterative, linearizedapproach. The resulting 3-D position accuracy is often de-scribed by the geometric dilution of precision, or GDOP [2].This measure, a function of GNSS constellation geometry(specifically the azimuths and elevation angles to the satellites

employed in the solution), is a condensed version of thecovariance matrix of the errors in the position and timeestimates. Combining the GDOP value and an estimate of theuser range error allows one to establish the 95% confidenceellipsoid.

For 2-D positioning on the surface of the earth, celestialnavigation is based upon measuring the altitudes (elevationangles, relative to an artificial or visible horizon, historicallymeasured with a sextant) of celestial bodies to locate theuser [1], [3]–[5]. Assuming multiple measurements of starelevations, finding ones’ position similarly involves solving anoverdetermined set of nonlinear equations; again, an iterative,linearized approach is known [6], [7].

In recent work these authors began to reexamine celestialnavigation from the vantage point of engineers conversant inGNSS language. Specifically, having developed lower boundsto GDOP for GNSS performance assessment in [8], wesimilarly considered the limits of performance of celestialnavigation by developing in [9] a lower bound to HDOP(horizontal dilution of precision), the natural metric to describethe impact of the geometry of the celestial bodies employed inthe position solution. As the development of this bound wasconstructive (meaning that we developed “balance” conditionson the azimuth of the stars employed in the position solution),we were also able to characterize what a good positioningconstellation might look like.

In GNSS a problem of interest is how to select a subsetof the available satellites for best performance (e.g. for a4 satellite fix it is common to choose those satellites thatform the tetrahedron with greatest volume [10]); a sizableamount of work has been done on this problem (see, forexample, [11]–[13]). Paralleling concepts from GNSS andcelestial navigation leads to the similar question of how toselect the stars for the best celestial fix. Over 200 years agoBowditch presented a basic solution for the star selectionproblem (which is still taught today!): “Choose the stars andplanets that give the best bearing spread . . . ” [1, p.276].This guidance is commonly interpreted as choosing stars thatprovide as uniform an azimuth distribution as possible (andnever, never select stars with nearly equal azimuths).

In our 2019 paper [9] we proved that this regular practiceis a sufficient, but not necessary, condition for best HDOPand is, in fact, overly restrictive. Specifically, we showed thatwhile azimuths forming the vertices of a regular polygon (i.e.a “uniform distribution”) meets the balance conditions, manyother selections of azimuths can achieve the balance conditions

as well (including some in which two or more stars are atthe same azimuth). For example, we showed that the union oftwo balanced patterns, each based upon a regular polygon, andthemselves with arbitrary rotation, also meets the conditions.This means that the class of possible optimum azimuths ismuch richer than Bowditch (or the typical mariner) imagines.In [9] these realizations resulted in a simple, yet effective, starselection algorithm.

In this current paper we continue to address questions andfacets of celestial navigation. Specifically, we consider twotopics: further advances in characterizing azimuth selection tominimize HDOP (providing some, perhaps, startling resultsfor today’s celestial users) and telling time from celestialobservations (clarifying the folklore on celestial estimates).These are expanded below.

II. CELESTIAL NAVIGATION

A. Review of the Basics

To the mariner, celestial navigation is the art of calculatinga 2-D position on the surface of the earth with the aid ofthe sun, moon, planets, and major stars. For aviators, it is thecalculation of a 2-D position relative to a known (barometric)altitude above the earth. In either case, it usually means thenavigator is obtaining lines of position (LOPs) by measuringthe altitudes of celestial bodies relative to an actual or artificialhorizon, typically by using a sextant. The general strategy incelestial navigation is to (1) choose a set of reasonable celestialbodies for measurement, based on visibility, azimuth, andaltitude, (2) measure observed altitudes of that set of celestialbodies at known time(s), (3) correct altitude measurements foreffects such as refraction, height of eye, or parallax, and (4)determine each celestial body’s location on the celestial sphereat the observation time (e.g. declination and Greenwich HourAngle). Each celestial body’s altitude measurement creates a“circle of equal altitude” on the surface of the earth. Theintersection of these circular lines of position results in aunique celestial fix. For a more local “small-scale” positionsolution, and with observed altitudes below 75 degrees, onecan draw these “circular arcs” as actual (straight) lines ofposition, perpendicular to the azimuth to the celestial object,and this is what is typically done in maritime navigation.

The classical method used to solve for position at seais called the “altitude-intercept” method, also known as theMarcq St. Hilaire method (after the French navigator whodeveloped the technique). In this method one uses the knowncelestial bodies’ locations plus an assumed position (AP),presumably reasonably close to the actual position, to calculatethe altitudes one would expect at known times at that AP. Thiscan be accomplished using the Nautical Almanac and Publica-tion 229 or https://aa.usno.navy.mil/data/docs/celnavtable.phpto determine both the computed altitudes and azimuth angles.The difference between the computed and observed altitudes(measured in minutes in arc), combined with the azimuth, isused to plot each celestial line of position. More specifically,the line of position produced is oriented perpendicular to theazimuth toward each celestial body, and is initially drawn

through the AP. The final LOP is shifted towards or away fromthe celestial body by the amount equal to the arc differencebetween the computed and observed altitudes (recognizing thatone minute of arc is approximately one nautical mile). If thecomputed altitude is greater than the observed altitude, weshift the LOP away from the celestial body along the azimuthline; if the computed altitude is less than the observed altitudewe shift the LOP towards the celestial body, along the azimuthline. This is done for each celestial body, and the final fix iscomputed from the intersection of all LOPs. If the lines do notintersect in a single point, the “estimation of the ship’s positionfrom the somewhat chaotic image of a number of position lines(is) left to the ‘keen judgement’ of the observer” [6].

B. HDOP

The lack of a common intersection of the LOPs in theoverdetermined case is familiar to those in the GNSS com-munity; the usual response is to reformulate the problemin a least squares sense, minimizing the residual error foreach LOP. Since the equations are nonlinear, this involveschoosing an initial assumed position, linearizing the LOPequations, solving for the least squares position, using thisresult as the next assumed position, and iterating. While thiscan be done in latitude/longitude coordinates [7], it is moreconvenient when describing positioning accuracy to employEast/North coordinates. Specifically, adapting the notation in[6], let δe and δn be the East and North least squares offsetsto the solution from the AP, respectively, and δk be thedifference between the measured and computed altitudes tothe celestial object at azimuth θk. Assuming m measurements,the simultaneous equations are

δk = δe sin θk + δn cos θk k = 1, 2, . . .m

We note that these are precisely the equations used in thealtitude-intercept method to specify the location of the LOPs.In other words the altitude-intercept method combined with the‘keen eye’ of the observer are equivalent to the first iterationin a typical least squares position solution.

While the least squares solution of this set of simultaneousequations is found in [6], it is more recognizable to the GNSScommunity to write these equations in vector form

δ1δ2...δm

=

sin θ1 cos θ1sin θ2 cos θ2

......

sin θm cos θm

[δeδn

](1)

orδ = G

[δeδn

]δ being the vector of altitude differences and G the directionscosine matrix

G =

sin θ1 cos θ1sin θ2 cos θ2

......

sin θm cos θm

=

e1 n1e2 n2...

...em nm

where we further simplify the notation using ek and nk as theEast and North components of the unit vector pointing towardazimuth θk. Assuming that the measurement errors are zeromean and uncorrelated with common variances σ2, the leastsquares solution to this overdetermined set of equations is[

δeδn

]=(GTG

)−1GT δ

and the covariance matrix of the resulting position error is

Cov([

δeδn

])= σ2

(GTG

)−1

Since the solution is a 2-D fix, on the surface of the Earth, itis common to focus on the Horizontal Dilution of Precisionor HDOP

HDOP =

√trace

{(GTG)

−1}

which is the square root of the sum of the variances in theEast and North directions without the measurement variancescaling. The fact that G has only two columns allows us tosimplify this performance measure. Specifically, since

GTG =

m∑k=1

e2k

m∑k=1

eknk

m∑k=1

eknk

m∑k=1

n2k

we have

HDOP =

√√√√√√m(

m∑k=1

e2k

)(m∑k=1

n2k

)−

(m∑k=1

eknk

)2 (2)

As an example consider the mariner observing the sky atthe twilight hours of Wednesday Jan. 30, 2019 somewhere tothe east of Reston, VA; say at 39N, 74W, about 30-40 milesoff of the coast from Wildwood, NJ. Using information fromHer Majesty’s Nautical Almanac Office and the US NavalObservatory, 29 catalogued stars are readily visible. Theirazimuths and elevations (rounded to the nearest whole degree)are:

azimuths =

1 16 25 31 64 7695 97 114 122 122 125125 135 172 178 205 209212 219 227 260 265 269309 319 322 332 360

and

elevations =

39 7 19 76 60 2754 15 36 40 10 5633 31 54 10 3 7226 53 3 36 56 1826 4 63 9 23

A sky plot showing these 29 locations appears in Figure 1.

113.2 177 204.6 218.9

226.5 268.2 308.3 318.2

Fig. 1. Sky plot for the example. The red and blue circles show the azimuthsand elevations of the full set of 29 star locations; the blue identifies the setof m = 8 stars with smallest HDOP (0.70711). The azimuths for this set areshown in the title.

Fig. 2. Histogram of the full search result. While the full range of values if[0.70711, 2.8020] we have truncated it on the right to show the sharp biasfor small HDOP values; only 361 of the 4,292,145 constellations have HDOPin excess of 1.4.

Imagine that our goal is to select m = 8 stars for de-termining our position. A common question is which 8 tochoose for best performance. With 29 available and m = 8,the problem is not too large; a full search requires testinga total of 4.3 million possible choices. Figure 2 shows theresulting histogram of the HDOP over all of the choices; thebest HDOP of 0.70711 through the worst case value of 2.8020(the constellation yielding the minimum HDOP is shown inblue in Figure 1). We note that this histogram is quite skewedtoward the left; there are very many 8-star constellations withexcellent performance!! Fully 50% have HDOP below 0.7195and 90% with HDOP below 0.752; hence, even choosing starsat random is likely to produce a pretty good result!

III. WHAT DO THE BEST CONSTELLATIONS LOOK LIKE?

Some relevant questions for celestial navigation include“What is the best possible performance (minimum HDOP)achievable?.” and “How should we select the celestial objectsemployed to achieve performance close to this best value?” In[9] we started answering these and related questions by firstdeveloping the lower bound

HDOP ≥√

4

m(3)

8 balance 4+4 balance 5+3 balance

Fig. 3. Skyviews of optimum 8 star configurations.

which is achieved by m celestial objects whose azimuthssatisfy two balance conditions

m∑k=1

eknk = 0 andm∑k=1

e2k =

m∑k=1

n2k =m

2(4)

Next, we started to characterize star constellations that metthese conditions. Specifically, we developed a series of results(proofs of these appear in [9]):

Theorem 1. Imagine that a set of azimuths {θ1, θ2, . . . , θm}is balanced. Then rotating the entire constellation by commonangle φ results in an equivalent balanced constellation. Sim-ilarly, negating all of the azimuths to {−θ1,−θ2, . . . ,−θm}yields another balanced constellation.

Since reflecting a constellation across any line through theorigin is equivalent to a rotation, negation, and rotation, wecan also form a balanced constellation via reflection. Further,since there are infinitely many choices for the rotation anglewe can form equivalence classes of constellations based upona rotation and reflection, with the need to describe only oneversion in each class.

Theorem 2. For m ≥ 3 the constellation forming a regularm-gon (azimuths evenly distributed over 360◦) is balanced.Without loss of generality the base constellation describingthis class has azimuths{

0◦,360

m, 2

360

m, 3

360

m, ..., (m− 1)

360

m

}We note that this solution certainly fits Bowditch’s notion ofbearing spread.

Theorem 3. The union of two balanced constellations yieldsanother balanced constellation.

Combining these three theorems we can build larger constel-lations from several smaller ones, each with its own arbitraryrotation; Figure 3 compares three such constellations:

• Left subfigure – 8 celestial objects each separated by 45◦

(drawn at common elevation for visual simplicity sinceHDOP is not a function of the elevations).

• Central subfigure – two sub-constellations of 4 celestialobjects. Each group can have an arbitrary rotation inazimuth; in fact, we could allow overlap meaning that aconstellation of pairs of stars with 90◦ azimuth separationwould meet the HDOP bound.

• Right subfigure – one set of 5 objects each separated by72◦ and the remaining 3 each separated by 120◦, eachwith arbitrary rotation.

For larger m there are more ways to construct union con-stellations (e.g. for m = 10 we could have 10 or 7 + 3 or6 + 4 or 5 + 5 or 4 + 3 + 3), so there are many classes ofbalanced constellations, all meeting the HDOP bound. Thetheory of partitions [14] can help in enumerating the numberof possibilities for larger m.

These prior results on balanced constellations can be ex-tended by three new theorems:

Theorem 4. Imagine that a set of azimuths {θ1, θ2, . . . , θm}is balanced. Then reflecting a single one of these azimuthsthough the origin, say θj to θj + 180◦, results in a balancedconstellation.

Proof. We check both conditions for the constellation witha single reflection; the method is primarily trigonometricmanipulations. Without loss of generality let j = 1. Usinghats to indicate the direction components of this modifiedconstellation, the first condition ism∑k=1

ek nk = sin(θ1 + 180◦) cos(θ1 + 180◦) +

m∑k=2

sin θk cos θk

= [− sin θ1] [− cos θ1] +

m∑k=2

sin θk cos θk

=

m∑k=1

sin θk cos θk =

m∑k=1

eknk

The last expression is the first balance condition for the

original constellation, equal to zero by assumption, so thiscondition is also zero for the constellation with a singlereflection. The second balance condition looks first at the sumof squares of the east components

m∑k=1

ek2 = sin2 (θ1 + 180◦) +

m∑k=2

sin2 θk

= [− sin θ1]2

+

m∑k=2

sin2 θk

= sin2 θ1 +

m∑k=2

sin2 θk

=

m∑k=1

sin2 θk =

m∑k=1

e2k

which is m2 by assumption. In the obvious, trivial way the sum

of the squares of the north components for the constellationwith one reflection is also m

2 and the result is a balancedconstellation.

This theorem naturally leads to several interesting classesof balanced constellations as follows:

Theorem 5. For m odd, the constellation class consisting ofazimuths

{0◦, 180m , 180·2m , . . . 180·(m−1)

m

}is balanced.

Proof. The proof is by construction. Start withthe regular m-gon constellation with azimuths{

0◦, 360m , 360·2m , . . . 360·(m−1)m

}of Theorem 2 and reflect

the m−12 largest of these (those greater than 180◦) back into

the range [0, 360]; the result is the desired uniform spacingof 180

m in azimuth.

Employing this theorem with m = 3 yields that a three pointconstellation with inter-azimuth spacing of 60◦ is balanced;with m = 5, five stars with 36◦ inter-azimuth spacing isbalanced.

In these authors’ opinion the most useful result of the singlereflection theorem is as follows:

Theorem 6. A pair of stars separated by 90◦ in azimuth isbalanced.

Proof. Following Theorem 4 this could be proved by con-struction using reflection (start with the balanced square con-stellation with azimuths {0◦, 90◦, 180◦, 270◦}, reflect the lasttwo yielding the balanced constellation with four azimuths{0◦, 0◦, 90◦, 90◦}, argue that this can be split this into twotwo-star sub constellations each of which must be balanced);however, it is more direct to just compute the balance condi-tions. Assuming azimuths of 0◦ and 90◦

2∑k=1

eknk = sin 0◦ cos 0◦ + sin 90◦ cos 90◦ = 0

2∑k=1

e2k = sin2 0◦ + sin2 90◦ = 1

2∑k=1

n2k = cos2 0◦ + cos2 90◦ = 1

and the two point constellation is balanced.

Sub-constellations of two stars with only a 90◦ separationallows for the inclusion of meteorological limitations to realstar siting; i.e. allowing for portions of the sky and/or horizonto be obscured (as occurs in a fog bank, when near land, orwith partial cloud cover). This is a significant improvementfrom Bowditch’s original recommendation of a uniform, 360degree spread in azimuth, allowing the mariner to optimizehis/her positioning accuracy even in limited sky conditions.

At this point we can attempt to characterize all classesof small sub-constellations (the statements below can all beshown to be true by simple, but tedious, algebraic operations):

• m = 2 – there is only one class, the pair with 90◦ spacing.• m = 3 – there are two classes, those with 120◦ spacing

and those with 60◦ spacing.• m = 4 – there are no new unique classes; all balancedm = 4 sub-constellations are formed from two m = 2selections each with 90◦ separation (but the two groupscan have any arbitrary relative rotation; hence, this is avery rich class).

• m = 5 – there are at least four classes: five azimuthswith 72◦ spacing, five with 36◦ spacing, and the twosub-sub constructions of 5 = 3 + 2. While it is possiblethat additional classes could exist, we have neither foundthem nor proved that they do not exist.

• m > 5 – we conjecture that the only interesting newclasses are when m is prime, providing two new classeswith 360◦

m and 180◦

m spacing, respectively.In summary, the union-based construction of a balanced con-stellation of Theorem 3 has a much, much larger set of buildingblocks then imagined in [9]; for example, optimum constel-lations for the m = 8 example above could be constructedas 5 + 3, 3 + 3 + 2, or 2 + 2 + 2 + 2. Further, we notethat the ability to have arbitrary rotations of each of the sub-constellations means that the optimum choices based uponthe decomposition of 8 = 2 + 2 + 2 + 2 by far dominatethe totality of optimum constellations (and we use this fact inthe algorithm development below). Combining this richness ofoptimum constellations with the relative insensitivity of HDOPto a small mismatch (also noticed in [9]) helps to explain theempirical fact (seen in Figure 2) that the histogram of HDOPperformance over all possible star selections is strongly biasedtoward good performance.

IV. A SIMPLE STAR SELECTION ALGORITHM

Recall that the goal is to select a subset of the visiblestars to yield small HDOP. In [9] we developed a simplesearch algorithm which looked for multiple small, but nearlybalanced sub-constellations with the intent of building a largerconstellation as the union of smaller ones. For the night skyexample shown in Figure 1 our best result using Theorems1-3 consisted of a pair of 4-tuples; the resulting HDOP was

4: 1 95 178 269 4: 31 125 219 309 4+4: 1 31 95 125 178 219 269 309

Fig. 4. Star selection from two 4-tuples; the resulting HDOP is 0.7075.

TABLE IAN EXAMPLE OF THE RESULTS OF THE PAIRWISE STAR ALGORITHM.

θ1 θ2 ∆θ

121.9◦ 211.9◦ 0.0◦

30.9◦ 121.0◦ 0.1◦

218.9◦ 308.3◦ 0.6◦

24.3◦ 113.2◦ 1.1◦

177.0◦ 268.2◦ 1.2◦

63.3◦ 331.8◦ 1.5◦

......

...

0.7075 (very close to the bound of 0.70711). Theorems 4-6provide for an alternative search method; specifically, searchfor pairs of stars separated by approximately 90◦ keeping them2 closest pairs. Specifically, the algorithm is:

1) For a total of N stars, compute the N(N − 1) azimuthdifferences (remember to take into account the modulo360◦ wraparound).

2) Sort the list by those closest to 90◦.3) Select the m

2 pairs at the top of the list.For the N = 29 azimuths listed above this sorted list isshown in Table 1 in which ∆θ denotes the offset from a 90◦

difference. The resulting constellation has HDOP = 0.70712;its sky plot is shown in Figure 5. While different then thesolution shown in Figure 1, it has nearly equal performance.

Of, perhaps, greatest utility of this pairwise algorithm is itsability to find excellent constellations even if a large sectionof the sky (as measured in azimuth) is unavailable as happenswhen a portion of the horizon is occluded either by a landmass or a fog bank (or other meteorological event). As anexample, consider the sky of the example above, but in whicha 100◦ swath of the NorthWest sky is obscured as suggestedby Figure 6 (azimuths between 240◦ and 340◦). The topsubfigure shows the resulting sky with the full search bestconstellation (HDOP = 0.70711); the bottom subfigure showsthe best pairwise constellation (HDOP = 0.70716). Figure 7shows an even more extreme example; both the brute forceand pairwise approaches find constellations with HDOP =0.70711 (interestingly, slightly better than the pairwise result

24.3 30.9 113.2 121

121.9 211.9 218.9 308.3

Fig. 5. Sky plot for the example. The red and blue circles show the azimuthsand elevations of the full set of 29 star locations; the blue identifies the bestset of m = 8 stars found using the pairwise search algorithm (with HDOP(0.7071). The azimuths for this set are shown in the title.

with more sky availability, but this just speaks to the slightsuboptimalty of the pairwise algorithm). It is pretty clear fromthe top subfigure in this case that the full search is also findingpatterns with pairwise separation of 90◦. While they are notidentical solutions, the performance of the fast algorithm isequivalent.

V. SOLVING FOR TIME

In GNSS we get time (i.e. an estimate of UTC) from thesolution algorithm; can we do the same with celestial? Websites such as that shown in Figure 8 suggest that you can.However, as we demonstrate below, these “clocks” yield localtime only (not traceable back to UTC, much like a sundial).In contrast, there is also the “Lunar Distance” method whichestimates precise time from the movement of the moon relativeto a star [16]. Below we examine estimating precise time fromcelestial measurements, demonstrating both of these facts and,further, presenting a way to predict the accuracy of any suchestimate.

Referring to [6], [7], [17]–[20] we start with the singlemeasurement equation for a celestial body

sinH = sinφ sin d+ cosφ cos d cos (G− θ)

in which φ and θ are, respectively, the user’s latitude andlongitude (for which we are trying to solve), G is the celestial

30.9 75.1 96 121.9

124.6 171.4 218.9 359.5

24.3 30.9 113.2 121

121.9 134.6 211.9 226.5

Fig. 6. Sky plots under the assumption of 100◦ of sky occlusion. The topsubfigure is the brute force best constellation while the bottom subfigure isthe best pairwise constellation.

113.2 121.9 124.2 124.6

204.6 208.5 211.9 218.9

113.2 121 121.9 134.6

204.6 208.5 211.9 226.5

Fig. 7. Sky plots under the assumption of 180◦ of sky occlusion. The topsubfigure is the brute force best constellation while the bottom subfigure isthe best pairwise constellation.

Fig. 8. A typical star clock website [15].

object’s Greenwich hour angle, d is the object’s declination(the latitude and longitude of the Ground Point under theobject, respectively, both of which are tabulated if the timeis known), and H is the altitude measurement. Solving for thealtitude

H = sin−1 {sinφ sin d+ cosφ cos d cos (G(t)− θ)}≡ g (φ, θ)

we can linearize the measurement equation about some nomi-nal point (with values H0, φ0, θ0, the δ’s denote the perturba-tions)

H0 + δh ≈ g (φ0, θ0) +∂g

∂φ(φ− φ0) +

∂g

∂θ(θ − θ0)

to yield a differential measurement equation

δh ≈ ∂g

∂φ

∣∣∣∣φ0,θ0

δφ+∂g

∂θ

∣∣∣∣φ0,θ0

δθ

with∂g

∂φ=

cosφ sin d− sinφ cos d cos (G− θ)√1− (sinφ sin d+ cosφ cos d cos (G− θ))2

and∂g

∂θ=

cosφ cos d sin (G− θ)√1− (sinφ sin d+ cosφ cos d cos (G− θ))2

If time, t, is unknown then the location of the celestialobject’s ground point, (d,G), should also vary with time, say(d(t), G(t)). For example, the Ground Point for stars on thecelestial sphere rotate daily in longitude at constant latitude,so a simple model for the stars is

d(t) = d0 and G(t) = G0 + κ (t− t0)

with κ approximately equal to 15 degrees per hour (actuallyslightly more at 360◦ per sidereal day). Objects within our

solar system also move relative to the Earth; hence, haveadditional motion beyond the rotation of the celestial sphere.Figure 9 shows the declination and GHA over a 24 hour periodfor the standard visible celestial objects (sun, moon, visibleplanets, and 58 tabulated stars). On this scale the declinationsappear to be constant and the GHAs to vary linearly. Figure10 shows the results of fitting straight lines to this data.From the top subfigure we observe that the stars all haveconstant declination (the ‘blue” slopes are all equal to zero),the Sun and planets have almost zero slopes (η ≈ 0), andonly the moon has a significant declination change. From thebottom subfigure we observe that the stars all have slope ofκ = 15.04 degrees per hour, the Sun and three inner planetshave somewhat different slopes, and the moon again showsthe largest difference. The quality of these fits suggest modelsof the form

d(t) = d0 + η (t− t0) and G(t) = G0 + κ (t− t0)

Further, we will add subscripts to index the different celestialobjects as needed. Adding in this time dependence we havethe linearized perturbation equation

δh ≈ ∂g

∂φ

∣∣∣∣φ0,θ0,t0

δφ+∂g

∂θ

∣∣∣∣φ0,θ0,t0

δθ +∂g

∂t

∣∣∣∣φ0,θ0,t0

δt

with

∂g

∂t=η (sinφ cos d− cosφ sin d cos (G− θ))

∆

− κ cosφ cos d sin (G− θ)∆

in which

∆ =

√1− (sinφ sin d+ cosφ cos d cos (G− θ))2

While this linearized model is interesting, it is more insightfulto recast it in a local coordinate frame (East, North, Up versuslatitude and longitude). We can accomplish this by a changeof variables.

First, starting with latitude φ, longitude θ, and height h(= 0) and, assuming a spherical Earth of radius re (= 3959miles), we can convert to ECEF (Earth Centered Earth Fixed)coordinates by

x = re cosφ cos θ

y = re cosφ sin θ

z = re sinφ

A small perturbation in latitude and longitude, φ → φ + δφand θ → θ + δθ, yields the corresponding perturbations inECEF

δx ≈ −re cos θ sinφ δφ− re sin θ cosφ δθ

δy ≈ −re sin θ sinφ δφ+ re cos θ cosφ δθ

δz ≈ re cosφ δφ

0 6 12 18 24

Time in hours

-100

-50

0

50

100

De

clin

atio

n in

de

gre

es

0 6 12 18 24

Time in hours

0

200

400

600

800

GH

A in

de

gre

es

Fig. 9. Declination and GHA over a 24 hour period for 58 tabulated stars (inblack) and five solar system objects (Sun, Moon, Mercury, Venus, and Mars;in red).

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05

Slope of d(t) in degrees per hour

-100

-50

0

50

100

Inte

rce

pt

Sun

Moon

Mercury

Venus

Mars

14.4 14.6 14.8 15 15.2

Slope of G(t) in degrees per hour

0

100

200

300

400

Inte

rce

pt

Sun

Moon

MercuryVenus

Mars

Fig. 10. Linear fits of the declination and GHA over a 24 hour period for 58tabulated stars (in blue) and five solar system objects (as labeled in red).

Next, the conversion from ECEF to ENU (East, North, Up)at the point (φ, θ, h) is a linear transformation

enu

=

− sin θ cos θ 0− sinφ cos θ − sinφ sin θ cosφcosφ cos θ cosφ sin θ sinφ

xyz

The differentials in this ENU frame are δeδnδu

=

− sin θ cos θ 0− sinφ cos θ − sinφ sin θ cosφcosφ cos θ cosφ sin θ sinφ

×

−re cos θ sinφ δφ− re sin θ cosφ δθ−re sinλ sinφ δφ+ re cos θ cosφ δθ

re cosφ δφ

=

re cosφ δθre δφ

0

so

δe ≈ re cosφ δθ and δn ≈ re δφ

orδφ ≈ δn

reand δθ ≈ δe

re cosφ

Returning to the perturbation model

δh ≈ ∂g

∂φ

∣∣∣∣φ0,θ0,t0

δφ+∂g

∂θ

∣∣∣∣φ0,θ0,t0

δθ +∂g

∂t

∣∣∣∣φ0,θ0,t0

δt

≈ cosφ0 sin d0 − sinφ0 cos d0 cos (G0 − θ0)

re ∆δn

+cos d0 sin (G0 − θ0)

re ∆δe

+η (sinφ0 cos d0 − cosφ0 sin d0 cos (G0 − θ0))

∆δt

− κ cosφ0 cos d0 sin (G0 − θ0)

∆δt

As noted above in Section II.B, it is usual in celestialnavigation to use the azimuth from the user’s position to-ward the celestial object’s Ground Point; this is the full360◦ arctangent with components cos d0 sin(G − θ0) andcosφ0 sin d0−sinφ0 cos d0 cos(G−θ0). Equivalently, the Eastand North components are proportional to the individual terms

east = sin az =cos0 φ sin d0 − sinφ0 cos d0 cos(G0 − θ0)

β

north = cos az =cos d0 sin(G0 − θ0)

β

in which β is the constant

β =

√(cosφ0 sin d0 − sinφ0 cos d0 cos(G0 − θ0))

2

+ (cos d0 sin(G0 − θ0))2

A tedious trigonometric argument shows that this is the sameas the square root term in the denominator of the Eastand North derivative expressions, β = ∆, so the linearizedequation becomes

δh ≈ sin az0re

δn+cos az0re

δe

+η (sinφ0 cos d0 − cosφ0 sin d0 cos (G0 − θ0))

∆δt

− κ cosφ0 cos az0 δt

Similarly, the first δt term’s coefficient can be related to whatwe’ll call the complementary azimuth, azc, the azimuth fromthe celestial object’s Ground Point back to the observer’sposition (note that in spherical trigonometry this is not, ingeneral, just a 180◦ rotation)

sinφ0 cos d0 − cosφ0 sin d0 cos (G0 − θ0)

∆= sin azc

so

δh ≈ sin az0re

δn+cos az0re

δe

+ (η sin azc − κ cosφ0 cos az0) δt

Finally, it is convenient to write these equations in unitsof distance. If δh is measured in arc minutes (as is typicalin celestial navigation), then multiplying by re yields units ofnautical miles (re · 2π

360·60 = 1 nm), so

δhnm ≈ sin az0 δn+ cos az0 δe+ re (η sin azc − κ cosφ0 cos az0) δt

Matching units, the right hand side must also be in unitsof nautical miles. Examining the first and second terms onthe right, the resulting δn and δe will each have units ofnautical miles since the sine and cosine terms are unitless.The remaining terms, reη sin azc δt and −reκ cosφ0 cos az0 δt,need some explanation. In Figure 10, η and κ are defined withunits of degree per hour, so the products have units of miles *degrees/hour * unit of δt. Multiplying by 2π

360 converts theseterms to

2π

360re (η sin azc − κ cosφ0 cos az0) δt

with units of miles * radians/hour * unit of δt. Multiplyingnow by 1

60 converts these last two terms to

2π

360 · 60re (η sin azc − κ cosφ0 cos az0) δt

with units of miles * radians/minute of time * unit of δt.However, as noted above, 2π

360·60re reduces to 1 nm, so theterms are

(η sin azc − κ cosφ0 cos az0) δt

with units of nautical miles per minute of time * unit of δt.In other words, the perturbation equation is of the form

δhnm ≈ sin az0 δnnm + cos az0 δenm+ (η sin azc − κ cosφ0 cos az0) δtmin

with δe and δn measured in nautical miles and δt measuredin minutes (hence, the subscripts).

In vector-matrix form, for m measurements we have

δh1δh2

...δhm

=

sin az1 cos az1 tterm1

sin az2 cos az2 tterm2

......

...

sin azm cos azm ttermm

δnδeδt

TABLE IIDOP VALUES UNDER VARIOUS CELESTIAL OBJECT CHOICES.

Time known Time unknown PositionHDOP known

Situation EDOP NDOP HDOP bound EDOP NDOP HDOP TDOP TDOPstars (16) only 0.35511 0.36025 0.50585 0.5000 ∞ 0.36222 ∞ ∞ 1.9804stars + Mercury 0.35272 0.34630 0.49431 0.48507 560.60 0.36034 560.60 3162.7 1.9532stars + Mercury + Venus 0.35270 0.32890 0.48230 0.47140 560.21 0.34483 560.21 3160.6 1.9498stars + Mercury + Venus + moon 0.34792 0.31215 0.46742 0.44721 37.601 0.32674 37.602 212.45 1.9402stars + moon (4 at 5 minute intervals) 0.34130 0.29600 0.45178 0.44721 21.562 0.36016 21.562 122.18 1.9341stars + moon (4 at 15 minute intervals) 0.33833 0.29809 0.45092 0.44721 20.140 0.35958 20.143 114.21 1.9185sun + moon (4 each at 15 minute intervals) 1.4707 0.43848 1.5347 0.70711 124.53 0.77620 124.53 752.13 7.5434moon only (4 at 5 minute intervals) 24.018 8.7772 25.631 1.0000 123170 1897.5 123190 726860 8.9956moon only (4 at 15 minute intervals) 7.6123 3.2783 8.2882 1.0000 12654 188.81 12655 74780 7.7341stars + moon (9 at 15 minute intervals) 0.31325 0.27028 0.41373 0.40000 13.530 0.35249 13.534 77.248 1.7549stars + moon (13 at 15 minute intervals) 0.28644 0.26635 0.39114 0.37139 10.804 0.33749 10.809 62.112 1.5774

in which we employ the notation

ttermk = ηk sin azc − κk cosφ0 cos az0

for the kth celestial object. This is the extension of (1) addingin the appropriate column for the unknown time (different, ofcourse, from the column of ones for the unknown clock biasin GNSS). An alternative, compact form is

δh = G δx

in which δx represents the desired position/time update vector.Solving for this update in a least squares sense

δx =(GTG

)−1GT δh

The covariance matrix of the resulting position/time error is

Cov (δx) = σ2(GTG

)−1

in which σ2 is the common variance of the assumed indepen-dent star altitudes; the diagonal elements of

(GTG

)−1are the

squares of the East DOP (EDOP), North DOP (NDOP), andtime DOP (TDOP), respectively. The full DOP is the squareroot of the trace of this covariance matrix

DOP =√

EDOP2 + NDOP2 + TDOP2

For comparison to the results above, the HDOP is the L2-normof the first two of these diagonal elements

HDOP =√

EDOP2 + NDOP2

Let’s consider this result:• First, imagine that all ηk = 0, i.e. that we are examining

only stars (i.e. not the moon or sun or any planets), sothat κk is a constant, say κ. The matrix G simplifies to

G =

sin az1 cos az1 −κ cosφ0 cos az1

sin az2 cos az2 −κ cosφ0 cos az2

......

...

sin azm cos azm −κ cosφ0 cos azm

and immediately we note that the second and thirdcolumns are linearly dependent; hence, the matrix G isrank deficient and the matrix equation is unsolvable. Theresult is that it is impossible to estimate both positionand time from only measurements of the stars (i.e. thelongitude and time variables are non-separable) and “starclocks” cannot provide precise time as was stated at thebeginning of this section.

• Now consider ηk 6= 0 for at least one celestial object. Ingeneral, G is now full rank, but the product GTG mightstill be poorly conditioned yielding high DOP. Table IIshows the results of computing DOP for a variety ofcelestial combinations as would be seen on June 1, 2020at a location off of the coast of Oregon:

– Each row shows the performance for three problems:(1) time known, but position unknown (the tradi-tional celestial navigation problem); (2) time andposition unknown (i.e. celestial navigation withouta clock); and (3) position known, time unknown (acelestial timing receiver).

– The first 4 rows assume that all of the celestialobjects’ elevations are measured simultaneously; theremainder assume sightings over time for the moonor sun (the stars are still just one simultaneousmeasurement). We focus a number of these examplesat moon sightings to show the value of these resultsto the evaluation of the performance of the LunarDistance method.

– The entry “stars” means that the 16 brightest starsare employed (Alioth, Arcturus, Bellatrix, Betel-geuse, Capella, Deneb, Denebola, Dubhe, Elnath,Kochab, Mirfak, Polaris, Pollux, Procyon, Regulus,and Schedar).

– In the case of known time, the resulting HDOP prettywell tracks the lower bound,

√4m , as long as the

stars are involved; situations with only the sun andmoon (or moon alone) have poor HDOP since thetime separated measurements of these objects donot provide much azimuth spread unless the timeseparations become large.

– When time is unknown we observe relatively goodNDOP whenever the 16 stars are involved (this is nottoo surprising in that Polaris by itself provides goodinformation on latitude without any knowledge oftime); the near inseparability of the estimates of timeand East manifests itself in the much larger valuesfor EDOP and TDOP.

– The very last row, stars and multiple moon measure-ments, shows that one can improve the accuracies ofthe East and time estimates, but that the performancewithout knowledge of time is still quite poor forthis significant amount of measurement data; hence,the limited movement of the natural solar system’sbodies makes for poor time estimation. It might beinteresting to see how man-made bodies, such as aspace station, might contribute to our ability to readtime from the sky.

– The final column considers the accuracy of a timeestimate assuming a precisely known position. Sincethe standard deviation of the altitude measurement istypically about 0.2 nm (0.2 minutes of arc), then thestandard deviation of the time estimate for many ofthe presented situations is approximately 25 seconds(0.2 ∗ 2 ∗ 60), about 2 minutes for the moon alone(which is quite close to the empirical accuracy ofthe Lunar Method mentioned in the literature [16]).While not stratum 1, still not too bad for such asimple method.

VI. CONCLUSIONS

The primary significance of this paper is that it corrects themisconception, starting with Bowditch, that the stars selectedin celestial navigation need to be “evenly” distributed across360 degrees of azimuth (and this same misconception existsfor GNSS with users thinking that the satellites should uni-formly cover the sky); that multiple stars from nearly similardirections (i.e. Polaris and stars in the Big Dipper) can bevery useful together in forming a good celestial fix. As asecondary contribution, we mathematically formulate what canand cannot be estimated about time from the observation ofcelestial objects.

It is obvious that the development could be extended to amore general noise model. Specifically, we assumed a commonvariance on the measurement noise for each equation; explic-itly writing a noise term εk for each altitude measurement

δk = sin θk δe + cos θk δn + ttimek δt+ εk

we have been operating under the assumption that E {εk} = 0and E

{ε2k}

= σ2. This could be extended as follows:

• Recognizing that each celestial object is of a differentbrightness, that lower elevation angles are subject to moreatmospheric uncertainty, and that parts of the sky and/orhorizon may be harder to see clearly, it would be morenatural to allow for unequal variances, E

{ε2k}

= σ2k.

Allowing for this variation, the natural solution is aweighted least squares[

δeδn

]=(GTΛ−1G

)−1GTΛ−1δ

with Λ = diag(σ21 , σ

22 , . . . , σ

2m

).

• The assumption of a zero mean for each error term can berelaxed somewhat to allowing for a common bias in thealtitude measurements (e.g. perhaps due to a systematicerror in the sextant) [21]; the result is to add a columnof ones to G, similar to the ones in the direction cosinesmatrix in GNSS, with the result that the least squaressolution also estimates this bias.

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