# GWDALI: A Fisher-matrix based software for gravitational wave parameter-
estimation beyond Gaussian approximation

Josiel Mendonça Soares de Souza  Riccardo Sturani

###### Abstract

We introduce GWDALI, a new Fisher-matrix, python based software that computes
likelihood gradients to forecast parameter-estimation precision of arbitrary
network of terrestrial gravitational wave detectors observing compact binary
coalescences. The main new feature with respect to analogous software is to
assess parameter uncertainties beyond Fisher-matrix approximation, using the
derivative approximation for Likelihood (DALI). The software makes optional
use of the LSC algorithm library LAL and the stochastic sampling algorithm
Bilby, which can be used to perform Monte-Carlo sampling of exact or
approximate likelihood functions. As an example we show comparison of
estimated precision measurement of selected astrophysical parameters for both
the actual likelihood, and for a variety of its derivative approximations,
which turn out particularly useful when the Fisher matrix is not invertible.

###### keywords:

Gravitational Waves , Derivative Approximation for LIkelihood , Fisher Matrix

[first]organization=Departamento de Física Teórica e Experimental, city=Natal,
state=RN, country=Brazil

[ufrn]organization=Instituto de Física Teórica, UNESP/ICTP-SAIFR, city=São
Paulo, state=SP, country=Brazil

###### Contents

  1. 1 Introduction
  2. 2 Astrophysical parameter estimation for coalescing binaries and Fisher matrix
    1. 2.1 Analytic Fisher Matrix Inversion
  3. 3 DALI algorithm for Gravitational Waves
    1. 3.1 Approximate likelihood: Doublet-DALI
  4. 4 GWDALI python package
    1. 4.1 Example of luminosity distance uncertainty
  5. 5 Conclusions
  6. A Derivative expansion beyond cubic order: P- and H-Tensors
  7. B Example of parameter recovery

##  1 Introduction

The advent of Gravitational Wave (GW) astronomy [1], with the detections made
by the LIGO [2] and Virgo [3] detectors, naturally calls for investigating
which physical quantity and how well it can be measured with the observations
made possible in this new gravitational channel.

The detector network should be enriched by the Japanese KAGRA [4] already in
the ongoing fourth science run of the so-called _second generation_ GW
detectors, and by the Indian Indigo [5] by the end of the current decade. For
the next decade, additional third generation detectors are foreseen, namely
the European Einstein Telescope (ET) [6] and the North-American Cosmic
Explorer (CE) [7].

While detections in the future may not be limited to compact binary
coalescences, they represent the totality of detections so far, and they are
expected to be the key observables for future observatories as well. It is
then natural to ask ourselves for expectations and forecast of precision
measures of astrophysical source parameters, how they depends on the intrinsic
detector feature, like spectral noise density, and localization and
orientation of the detectors composing the network.

Parameter estimation is naturally performed within a Bayesian inference
framework, which allows determination of most likely values of signal model
parameter as well as their confidence interval. A common tool for rapid
precision forecast is the _Fisher matrix_ approximation, which consists in
approximating the logarithm of the likelihood in the proximity of its maximum
value with a Taylor expansion truncated at quadratic order. By inverting the
matrix of second derivatives of the log-likelihood with respect to all
parameters one can then obtain the _covariance_ matrix, which gives direct
assessment of marginalized uncertainties of physical parameters [8]. On one
hand the Fisher matrix approximation is ubiquitous in data analyses, as an
invaluable tool to provide analytic and easy-to-compute estimates of parameter
uncertainties, on the other hand there are cases in which it can fail, e.g. in
presence of approximately flat directions and/or correlations among
parameters, giving rise to zero determinant second derivative matrix.

In this case it is natural to investigate how higher order approximations may
perform with respect to the quadratic one and/or the complete likelihood. The
goal of the present work is to introduce a tool to perform a suitable Taylor
expansion of the derivative approximation of the log-likelihood. This is
implemented in the publicly available python code
GWDALI111github.com/jmsdsouzaPhD/GWDALI/, complementing standard Fisher matrix
codes already available in literature like GWBENCH [9], GWFISH [10], and
GWFAST [11].

In particular we borrow here the idea to use higher (than second) derivative
approximations to the log-likelihood from [12], where the Derivative
Approximation for LIkelihood (DALI) was first proposed in a general context
and applied to cosmological examples, and from [13], where a first example of
DALI applied to GW source parameter inference has been proposed.

The paper is structured as follows: in Sec. 2 we give an overview of parameter
estimation for GW signal from coalescing binaries highlighting some known
limit of the Fisher-matrix approximation, followed in Sec. 3 by a description
of the DALI method. In Sec. 4 we give an example of the benefits of using
GWDALI for estimating parameter uncertanties and finally in Sec. 5 we outline
possible future applications and development of the GWDALI software.

##  2 Astrophysical parameter estimation for coalescing binaries and Fisher
matrix

The coalescence of compact binary systems can be divided in three phases:
inspiral, merger and ring-down. In particular the inspiral phase admits an
analytic description in terms of approximations to General Relativity, the
most successful of which in building waveform models has been the post-
Newtonian approximation [14, 15]. The subsequent merger phase is in principle
determined by the inspiral one, of which it is the continuation, and it has
been described solved with numerical methods [16]. Finally the ring-down phase
also admits an analytic description, in terms of damped exponentials, and
several phenomenological models exist nowadays providing complete analytic
waveforms encompassing the three phases, see e.g. [17, 18]. An exhaustive set
of waveform approximant is available in the LSC algorithm library (LAL) [19],
which are callable by python or C codes.

The binary systems of interest for current detectors are expected to be in
circular orbits, as angular momentum is more efficiently radiated than energy.
Orbits are indeed expected to circularize in the early stage of inspiral [20],
leaving little or no eccentricity when reaching the LIGO/Virgo band
($10-10^{4}$ Hz), which is not too distant from the merger frequency $f_{m}$,
which can be roughly approximated by $(2\pi M)^{-1}$, being $M$ the total mass
of the binary system, obtaining $f_{m}\sim 32$kHz$(M/M_{\odot})^{-1}$.222We
use “geometric units”, with $G=1=c$. The astrophysical parameters defining a
circular binary coalescences are 15 [21], organized in two categories [22]
according to if they affect the morphology of the waveform (intrinsic), or
just its overall shape (extrinsic). The 15 parameters are summarized in Tab.
1.

Intrinsic parameters | Extrinsic parameters
---|---
$M_{c}$, $\eta$, $\vec{S}_{1}$, $\vec{S}_{2}$ |  $d_{L}$, $\psi$, $\iota$, $\phi$, $\alpha$, $\beta$, $t$
Table 1: Parameters defining the observation of a binary system observation
(whose constituents are treated as point-like object), divided between
intrinsic ($m_{1,2}$ are the constituent masses, $\vec{S}_{1,2}$ are the
binary constituent spin vectors) and extrinsic (see text).

Given the time series of the output of the $i$-th detector $d_{i}$ and a
_template_ signal $h_{t_{i}}$, the log-likelihood for astrophysical parameter
inference is proportional the norm-squared $||d_{i}-h_{t_{i}}||^{2}$, where
the norm is inherited from the scalar product

$\langle
h_{1},h_{2}\rangle=2\int_{0}^{\infty}\frac{\tilde{h}_{1}(f)\tilde{h}_{2}^{*}(f)+\tilde{h}_{1}^{*}(f)\tilde{h}_{2}(f)}{S_{n}(f)}df\,,$
(1)

where $S_{n}(f)$ is the one sided _spectral noise density_ of the detector
defined by averaging the noisy detector output:

$\displaystyle\langle\tilde{n}(f)\tilde{n}^{*}(f^{\prime})\rangle\equiv\frac{1}{2}S_{n}(f)\delta(f-f^{\prime})\,.$
(2)

The Einstein Telescope (ET) [6] and Cosmic Explorer (CE) [23] sensitivity
curves adopted in this work are shown in Fig. 1.

The detector response to a GW is a linear combination of the two polarizations
with weights given by the _pattern functions_ $F_{+,\times}$, which depend on
sky localization of the source (defined by polar angles $\alpha,\beta$) and
the polarization angle $\psi$:

$h_{d_{i}}=F_{+}(\alpha,\beta,\psi)h_{+}+F_{\times}(\alpha,\beta,\psi)h_{\times}\,.$
(3)

The specific form of the pattern functions depend on the type of detector
considered. For interferometric detectors with opening angle $\Omega$ they
are, see ch. 7 of [24],

$\displaystyle\begin{array}[]{rcl}F_{+}&=&f_{+}\cos(2\psi)+f_{\times}\sin(2\psi)\,,\\\
F_{\times}&=&f_{\times}\cos(2\psi)-f_{+}\sin(2\psi)\,,\end{array}$ (6)

with

$\displaystyle\begin{array}[]{rcl}f_{+}&\equiv&\sin\Omega\frac{1}{2}(1+\cos^{2}\beta)\cos(2\alpha)\,,\\\
f_{\times}&\equiv&-\sin\Omega\cos\beta\sin(2\alpha)\,.\end{array}$ (9)

In the lowest order approximation (quadrupole formula) the polarizations for
the inspiral phase admit a simple analytic form, see e.g. ch.4 of [24]

$\displaystyle\begin{array}[]{rcl}\displaystyle\tilde{h}_{+}(f)&=&\displaystyle\frac{1+\cos^{2}\iota}{2}\pi^{-2/3}\left(\frac{5}{24}\right)^{1/2}\frac{\mathcal{M}_{c}^{5/6}f^{-7/6}}{d_{L}}e^{i\Phi(f)}\,,\\\
\displaystyle\tilde{h}_{\times}(f)&=&\displaystyle\cos\iota\,\pi^{-2/3}\left(\frac{5}{24}\right)^{1/2}\frac{\mathcal{M}_{c}^{5/6}f^{-7/6}}{d_{L}}ie^{i\Phi(f)}\,,\end{array}$
(12)

where $\mathcal{M}_{c}$ is the redshifted _chirp mass_ defined by
$\mathcal{M}_{c}\equiv\eta^{3/5}M(1+z)$, with $M$ the _intrinsic_ total mass
$M\equiv m_{1}+m_{2}$, $\eta$ the symmetric mass ratio $\eta\equiv
m_{1}m_{2}/M^{2}$, $z$ is the redshift and $\Phi(f)$ has an expansion known up
to 4th perturbative order in terms of the PN expansion parameter
$v^{2}\equiv(\pi Mf)^{2/3}$, see e.g. [25] and depends on $\eta$ and linearly
on the arrival time $t$ and the phase $\phi$. Note that $\iota,\phi,\psi$
compose the three Euler angles defining the orientation of the source axis
triad with respect to the observation one [26].

The log-likelihood vanishes when the template matches exactly the data, the
Fisher matrix for the $i$-th detector being

$\displaystyle{\cal F}_{i,ab}=4{\cal
R}e\int_{0}^{\infty}\frac{\partial_{\theta_{a}}\tilde{h}_{t_{i}}\partial_{\theta_{b}}\tilde{h}_{t_{i}}^{*}}{S_{n}(f)}df\,,$
(13)

where dependence of $h_{t_{i}}$ (and $\tilde{h}_{t_{i}}$) upon parameters
$\theta$ is understood in the notation. The $k$-th element of the diagonal of
the inverse of the Fisher matrix, the _covariance_ matrix, returns
marginalized uncertainties for the $k$-th variable, which then crucially
depend on correlation among parameters.

One of the obvious shortcomings of the Fisher matrix approximation is
encountered when the Fisher matrix itself is not invertible, hence it is not
possible to find the covariance matrix. This is exemplified in Fig. 2, where
the result for $d_{L}$ of the four dimensional covariance matrix in
$\\{d_{L},\iota,\psi,\phi\\}$ are displayed, as obtained with the code GWFISH
for the ET detector (made of three interferometers arranged in a equilater
triangles) and for CE (one $L$-shaped interferometer).

Figure 1: ET [27] and CE [28] noise spectral densities.

Figure 2: Example of $\Delta d_{L}/d_{L}$ vs. $\iota$ for a 4-dimensional
Fisher matrix in the parameters: $\\{d_{L},\iota,\psi,\phi\\}$ from GWFISH
code (blue line) [10], full likelihood for the same parameters (black) and for
the analytic curve given by $100/SNR$ (red). GW signals are generated by equal
mass spin-less binaries with $M=3M_{\odot}$ with TaylorF2 approximant [29],
averaged over sky position and inclination angles for ET (left) and CE
(right).

The increase of $d_{L}$ uncertainty, which is unbound for $\iota\to 0,\pi$ in
the Fisher matrix approximation, is due to $det({\cal F})\to 0$ as $\iota\to
0,\pi$, signaling a degenerate direction in the parameter space, and
invalidating the Gaussian approximation the Fisher matrix approach is based
on. The fact that this is a spurious behaviour due to the Fisher approximation
of the exact likelihood, is highlighted by the analaog quantity obtained by
MonteCarlo sampling of the likelihood, black line in Fig. 2. We discuss in
Sec. 3 a proposal about how to handle this problem.

###  2.1 Analytic Fisher Matrix Inversion

The inverse of the Fisher matrix is the covariance matrix, whose diagonal
entries give the fully marginalized 1$\sigma$ errors (in the Gaussian
approximation) of the corresponding parameters, which is probably the most
useful property of the Fisher method. The diagonal entries of the covariance
matrix are different depending on how many parameter one keeps fixed to the
maximum likelihood values, i.e. they depend on the dimensionality of the
Fisher matrix. As an example, the $d_{L}$ 1$\sigma$-uncertainty with all other
parameters fixed can be computed as:

$\left(\frac{\Delta
d_{L}}{d_{L}}\right)^{2}=\frac{1}{SNR^{2}}=\frac{4||k||^{-2}}{\left[(1+\cos^{2}\iota)^{2}F_{+}^{2}+4cos^{2}\iota
F_{\times}^{2}\right]}\,,$ (14)

where using the inspiral expressions (12) one has

$||k||^{2}\equiv\frac{5}{6}\frac{(G\mathcal{M}_{c})^{5/3}}{\pi^{4/3}d_{L}^{2}}\int_{0}^{\infty}\frac{f^{-7/3}}{S_{n}(f)}df\,,$
(15)

and the $SNR$ is defined as the norm of the GW signal $h$ given the scalar
product in eq. (1).

Marginalizing over $\iota$, instead of pinning it to its maximum likelihood
value, shows the effect of correlations into determining parameter
uncertainty. In this case, one can then read the $d_{L}$ uncertainty as

$\left(\frac{\Delta
d_{L}}{d_{L}}\right)^{2}=\frac{Cov(d_{L},d_{L})}{d_{L}^{2}}=\frac{\langle\partial_{\iota}h,\partial_{\iota}h\rangle}{d_{L}^{2}det({\cal
F})}=||k||^{-2}\frac{4(\cos^{2}\iota
F_{+}^{2}+F_{\times}^{2})}{\sin^{4}\iota(F_{+}F_{\times})^{2}}\,.$ (16)

This is a simple example of how correlations among parameters can degrade the
precision of individual parameters. In particular Figs. 3 report the analytic
Fisher matrix prediction of the uncertainty estimate of $d_{L}$ as a function
of the injected value of $\iota$ in two cases: 1-dimensional Fisher matrix and
bi-dimensional for parameters $d_{L},\iota$ (sources are located above a CE-
like detector).

Figure 3: 1$\sigma$ relative uncertainty in $d_{L}$ fixing all other
parameters (dashed red) or marginalizing over $\iota$ (solid black).

The presence of a zero eigenvalue in the Fisher matrix for $\iota\to 0,\pi$
denotes the existence of a flat direction, invalidating the Gaussian
approximation. In this example such breakdown manifests itself by the
divergence of the Fisher matrix prediction of $\Delta d_{L}$.

Note that standard recipes for obtaining the (pseudo-)inverse of a zero-
determinant Fisher matrix are adopted in e.g. GWFISH [10] and GWFAST [11],
where the Penrose method for generalized inverse [30] is implemented. Such
method consists of excluding from the inversion the subspace corresponding to
the kernel of the Fisher matrix. However identifying a zero eigenvalue
numerically depends on an arbitrary threshold.333In GWFISH the threshold for a
”numerical zero” is set to $10^{-10}$. We propose in the next section the
application of higher derivative likelihood approximation to GW parameter
estimation.

##  3 DALI algorithm for Gravitational Waves

Following the introduction of higher derivative in the likelihood expansion in
[12], we now introduce the GWDALI implementation of likelihood beyond Gaussian
order. Let $\theta$ be the set of GW parameters listed in Tab. 1. The Fisher
matrix approximation relies on a Taylor expansion of the log-likelihood around
the maximum likelihood value. Given the definition (1), the log-likelihood as
the norm-squared of the difference between data and template, one has the
straightforward Taylor expansion beyond quadratic order

$\displaystyle\begin{array}[]{rcl}\log\mathcal{L}&=&\displaystyle-\frac{1}{2}||d_{i}-h_{t_{i}}(\theta)||^{2}\\\
&\simeq&\displaystyle\log\mathcal{L}_{0}+\cancel{\sum_{a}\left(\partial_{a}\log\mathcal{L}\right)_{0}\Delta\theta_{a}}+\frac{1}{2!}\sum_{a,b}\left(\partial_{a}\partial_{b}\log\mathcal{L}\right)_{0}\Delta\theta^{ab}\\\
&&\displaystyle+\frac{1}{3!}\sum_{a,b,c}\left(\partial_{a}\partial_{b}\partial_{c}\log\mathcal{L}\right)_{0}\Delta\theta^{abc}\\\
&&\displaystyle+\frac{1}{4!}\sum_{a,b,c,d}\left(\partial_{a}\partial_{b}\partial_{c}\partial_{d}\log\mathcal{L}\right)_{0}\Delta\theta^{abcd}\\\
&&\displaystyle+\frac{1}{5!}\sum_{a,b,c,d,e}\left(\partial_{a}\partial_{b}\partial_{c}\partial_{d}\partial_{e}\log\mathcal{L}\right)_{0}\Delta\theta^{abcde}+\ldots\,,\end{array}$
(22)

where $\Delta\theta^{a}\equiv\theta^{a}-\theta^{a}_{0}$,
$\Delta\theta^{a_{1}\dots
a_{n}}\equiv\Delta\theta^{a_{1}}\dots\Delta\theta^{a_{n}}$ and the subscript
“$0$” stand for maximum likelihood value of the parameters. The first order
term of the Taylor expansion has been slashed out as it vanishes.

One can then define derivative tensors beyond the Fisher matrix

$\displaystyle\begin{array}[]{ll}\text{Fisher Matrix: }{\cal
F}_{ab}&\displaystyle\equiv-\left(\partial_{a}\partial_{b}\log\mathcal{L}\right)_{0}=4\sum_{i=1}^{N_{det}}{\cal
R}e\int_{0}^{\infty}\left(\frac{\partial\tilde{h}_{t_{i}}}{\partial\theta^{a}}\frac{\partial\tilde{h}_{t_{i}}}{\partial\theta^{b}}\right)\frac{df}{S_{n}(f)}\\\
&\displaystyle\equiv\left\langle\partial_{a}h|\partial_{b}h\right\rangle\,,\\\
\text{Flexion Tensor: }{\cal
S}_{abc}&\displaystyle\equiv-\left(\partial_{a}\partial_{b}\partial_{c}\log\mathcal{L}\right)_{0}\,,\\\
\text{Quarxion Tensor: }{\cal
Q}_{abcd}&\displaystyle\equiv-\left(\partial_{a}\partial_{b}\partial_{c}\partial_{d}\log\mathcal{L}\right)_{0}\,,\\\
\text{P-Tensor:
}\mathcal{P}_{abcde}&\displaystyle\equiv-\left(\partial_{a}\partial_{b}\partial_{c}\partial_{d}\partial_{e}\log\mathcal{L}\right)_{0}\,,\\\
\text{H-Tensor:
}\mathcal{H}_{abcdef}&\displaystyle\equiv-\left(\partial_{a}\partial_{b}\partial_{c}\partial_{d}\partial_{e}\partial_{f}\log\mathcal{L}\right)_{0}\,.\end{array}$
(29)

The explicit expression of the symmetric derivative tensors at third order
Taylor expansion ($O(\Delta\theta^{3})$), or Flexion term, is given by the
addition of 3 terms that can be arranged as follows:

$\displaystyle{\cal
S}_{abc}=\left\langle\partial_{a}h|\partial_{b}\partial_{c}h\right\rangle_{0}+(a\leftrightarrow
b)+(a\leftrightarrow c)\,,$ (30)

and its contribution to the likelihood is given by

$\displaystyle\begin{array}[]{cl}\log\mathcal{L}_{3}&\displaystyle\equiv-\frac{1}{6}\sum_{a,b,c}{\cal
S}_{abc}\Delta\theta^{abc}\\\
&\displaystyle=-\frac{1}{2}\sum_{a,b,c}\left\langle\partial_{a}h|\partial_{b}\partial_{c}h\right\rangle_{0}\Delta\theta^{abc}\,,\end{array}$
(33)

where at most second derivative of the waveform are involved. Note that an
approximate log-likelihood truncated at $O(\Delta\theta^{3})$ is not negative
semi-definite, causing a fundamental problem in the probabilistic
interpretation of the likelihood.

However, as explained in [12], this problem is fixed by truncating the
expansion in number of derivatives, rather than in powers of $\Delta\theta$.
Going beyond Gaussian level then requires to collect all terms involving a
given number of derivatives. To have a consistent expansion with two
derivatives one needs to include the relevant part of the Quarxion term:

$\displaystyle\begin{array}[]{cl}{\cal
Q}_{abcd}=&\displaystyle\left[\left\langle\partial_{a}\partial_{b}h|\partial_{c}\partial_{d}h\right\rangle+(b\leftrightarrow
c)+(b\leftrightarrow d)\right]\\\
&+\left[\left\langle\partial_{a}h|\partial_{b}\partial_{c}\partial_{d}h\right\rangle_{0}+(a\leftrightarrow
b)+(a\leftrightarrow c)+(a\leftrightarrow d)\right]\,,\end{array}$ (36)

where the terms in the last line involve the maximum number of derivatives (3)
at this $\Delta\theta$-expansion order (4).

The Quarxion contribution to the likelihood is

$\displaystyle\begin{array}[]{cl}\log\mathcal{L}_{4}&\displaystyle\equiv-\frac{1}{24}\sum_{abcd}{\cal
Q}_{abcd}\Delta\theta^{abcd}\\\
&\displaystyle=-\sum_{a,b,c,d}\left[\frac{1}{8}\left\langle\partial_{a}\partial_{b}h|\partial_{c}\partial_{d}h\right\rangle_{0}+\frac{1}{6}\left\langle\partial_{a}h|\partial_{b}\partial_{c}\partial_{d}h\right\rangle_{0}\right]\Delta\theta^{abcd}\,.\end{array}$
(39)

Only the first term of the square bracket in the second line of (39)
contributes to the two derivative expansion term, while the remaining one
contributes to the three derivative expansion terms, which is treated in A.

###  3.1 Approximate likelihood: Doublet-DALI

The Doublet-DALI is obtained from Taylor expansion of the log-likelihood and
collecting terms involving up to 2nd-order derivatives, hence it is obtained
by adding the Flexion term and the two-derivative part of the Quarxion term.
The Triplet-DALI is obtained by including terms with up to 3rd-order
derivatives, which involve parts of the Quarxion, the P- and the H-Tensor,
which are treated in A.

The complete result for the Doublet-DALI is

$\displaystyle\begin{array}[]{rl}\displaystyle\log\mathcal{L}_{Doublet}=&\displaystyle\log\mathcal{L}_{0}-\frac{1}{2}\sum_{a,b}\left\langle\partial_{a}h|\partial_{b}h\right\rangle_{0}\Delta\theta^{ab}\\\
&\displaystyle-\frac{1}{2}\sum_{a,b,c}\left\langle\partial_{a}h|\partial_{b}\partial_{c}h\right\rangle_{0}\Delta\theta^{abc}-\frac{1}{8}\sum_{a,b,c,d}\left\langle\partial_{a}\partial_{b}h|\partial_{c}\partial_{d}h\right\rangle_{0}\Delta\theta^{abcd}\,.\end{array}$
(41)

##  4 GWDALI python package

The main call to the GWDALI package can be exemplified as follows:

⬇

1Result = GWDALI(Detection_Dict,

2 FreeParams,

3 detectors,

4 approximant = ’TaylorF2’,

5 fmin = 1.,

6 fmax = 1.e4,

7 fsize = 3000,

8 dali_method = ’Doublet’,

9 sampler_method = ’nestle’,

10 npoints = 100,

11 save_samples = False,

12 save_cov = False,

13 save_fisher = False,

14 plot_corner = False,

15 hide_info = False,

16 rcond = 1.e-4,

17 index = 1)

18

19# np: length of FreeParams

20# Result[’Fisher’] : Fisher Matrix >> shape: (np,np)

21# Result[’CovFisher’] : Covariance from Fisher (inversion) >> shape: (np,np)

22# Result[’Covariance’] : Covariance from DALI >> shape: (np,np)

23# Result[’Samples’] : samples >> np columns

24# Result[’Error’] : list of 1-sigma uncertainties of parameters

25# Result[’Recovery’] : list of recovered params

The `detection_dict` input parameter is a dictionary containing all injection
parameter names and values. The `detectors` input contains all information
about detector (location on earth, opening angle between arms, spectral noise
density). The user defines via the dictionary `FreeParams` the variables for
which marginalized uncertainties have to be computed. In the case of a beyond
Fisher matrix approximation of the likelihood, i.e. Doublet or Triplet, a
MonteCarlo sampling is run to estimate the uncertainties and the maximum
likelihood estimators of the free parameters. A Bayesian inference code with
sampler method specified in `sampler_method` is run via Bilby [31] to perform
a search over the parameters specified in `FreeParams`.

Some features of the GWDALI software that may be worth emphasising are:

  * 1.

The user can choose among the methods: Fisher, Fisher_Sampling , Doublet or
Triplet for the Likelihood derivative approximation, the last three involving
a stochastic sampler to compute the (marginalized) likelihood, while the first
one is analytic in the numerical expression of the waveform derivatives.

  * 2.

User defined detector location coordinate, orientation and geometry.

  * 3.

The GW waveform can be generated via a LAL call, hence all approximants and
features available in LAL can be used .

  * 4.

The code returns MonteCarlo samples, Fisher and covariance matrix, as well as
the SNR.

In Figs.4,5 we report examples of 2-dimensional posteriors for the parameter
pair $\\{d_{L},\iota\\}$ for the two detectors whose spectral noise density
are reported in Fig.1.

Figure 4: Example of Bayesian inference results obtained with one Einstein
Telescope (triangle-shaped) detector, searching over parameters
$\\{d_{L},\iota\\}$, obtained with GWDALI.

Figure 5: Example of Bayesian inference results obtained with one Cosmic
Explorer ($L$-shaped) detector, searching over parameters $\\{d_{L},\iota\\}$,
obtained with GWDALI.

###  4.1 Example of luminosity distance uncertainty

As an example, we run GWDALI with 3 free parameters $\\{d_{L},\iota,\psi\\}$
and we compare the result of the Fisher matrix approximation with those of
higher derivative approximations of the likelihood and with the outcome of a
Bayesian inference code run on the exact likelihood. We report examples for 3
different GW-injection distances, corresponding to redshifts
$\\{0.1,0.5,1\\}$.

Figs.6,7,8 report the relative uncertainties in $d_{L}$, each figure
consisting of two plots: the one on the left is for a triangle-shaped detector
as ET, the one on the right for a $L$-shaped one as for CE, with spectral
noise sensitivities as in Fig. 1.

One can see the "turn around" in the Fisher matrix derived uncertainty of
$d_{L}$, which overall grows for $\iota\to 0,\pi$ until reaching a maximum
value and then decreasing. This is due to the onset of the above mentioned
Penrose method, which prevents taking the inverse of the full matrix when the
minimum eigenvalue (in absolute value) goes below a threshold, which in GWDALI
has been fixed to a the moderate value of $10^{-4}$ (`rcond` parameter shown
in the example code above). For minimum eigenvalues (in absolute value) lower
than the threshold, the Fisher matrix is effectively treated as lower
dimensional. In the ET case, the Fisher matrix has no vanishing eigenvalue
unless $\iota=0,\pi$, where it gains 1 zero eigenvalue. The $d_{L}$
uncertainty grows for $\iota\to 0,\pi$ until the Penrose pseudo-inverse method
excludes the flat direction by effectively reducing the dimensionality of the
Fisher matrix.

In the ET case one has 1 zero eigenvalue for _any_ value of $\iota$, hence the
flat direction is always chopped out.

On the other hand the Doublet and Triplet approximations do not present
pathological behaviour, broadly following the full-likelihood result. For
completeness, in B we report the relative difference between the innjected
value of $d_{L}$ and its maximum likelihood value for methods involving a
MonteCarlo sampling of the (exact or approximate) likelihood, showing
comparable behaviour of the Doublet and Triplet approximation and of the exact
likelihood.

Figure 6: Estimates of $\Delta d_{L}/d_{L}$ vs. $\iota$ averaged over 300
simulations for equal mass, spin-less binary system of fixed total mass
$3M_{\odot}$, $d_{L}\simeq 460$ Mpc corresponding to redshift $z=0.1$ in the
standard $\Lambda CDM$ model with $H_{0}=70\,km/s/Mpc$ and $\Omega_{M}=0.3$.
Cosmological computation makes use of astropy [32].

Figure 7: Same as in Fig. 6, but with redshift $z=0.5$, or $d_{L}\simeq 2.8$
Gpc.

Figure 8: Same as in Fig. 6, but with redshift $z=1$, or $d_{L}\simeq 6.6$
Gpc.

##  5 Conclusions

We have presented a novel python package for gravitational wave parameter
uncertainty forecast based on extending the log-likelihood derivative
approximation to higher level than two derivatives. The main scope is to avoid
large, unphysical uncertainty forecast for parameters when the Fisher matrix
is not invertible.

This software is highly flexible as it allows to call the LAL waveform
generator and the stochastic sampling is performed via Bilby. It allows to
select an arbitrary detector network by defining detector locations,
orientations and topologies.

A crucial step in the calculation of approximate likelihood is derivative
computation, so making the precision of numerical derivatives as high as
possible is crucial to improve the consistency of the code, which we tested by
comparison with results obtained by MonteCarlo sampling of the exact
likelihood.

Natural extensions of this code include cosmological applications [33, 34]
with assessment of cosmological parameter precision when studying population
instead of individual detections.

##  Appendix A Derivative expansion beyond cubic order: P- and H-Tensors

The Taylor expansion of the likelihood (22) has a 4-th order term specified by
the P-Tensor (29):

$\displaystyle\mathcal{P}_{ijklm}$
$\displaystyle\equiv-\left(\partial_{i}\partial_{j}\partial_{k}\partial_{l}\partial_{m}log\mathcal{L}\right)_{0}$
$\displaystyle=-\left[\left\langle\partial_{l}\partial_{m}h|\partial_{i}\partial_{j}\partial_{k}h\right\rangle+(9\
\rm{perms})\right]-\left[\left\langle\partial_{i}h|\partial_{j}\partial_{k}\partial_{l}\partial_{m}h\right\rangle+(4\
{\rm perms})\right]$

whose contribution to the likelihood is

$\displaystyle log\mathcal{L}_{5}$
$\displaystyle=-\frac{1}{5!}\sum_{a,\cdots,e}\mathcal{P}_{abcde}\Delta\theta^{abcde}$
$\displaystyle=-\sum_{a,\cdots,e}\left[\frac{1}{12}\left\langle\partial_{a}\partial_{b}h|\partial_{c}\partial_{d}\partial_{e}h\right\rangle+\frac{1}{24}\left\langle\partial_{a}h|\partial_{b}\partial_{c}\partial_{d}\partial_{e}h\right\rangle\right]\Delta\theta^{abcde}\,.$

It contains both 3- and 4- derivative terms, only the former contributing to
the Triplet-DALI.

The $H$-Tensor term is

$\displaystyle\mathcal{H}_{abcdef}\equiv$
$\displaystyle-\left(\partial_{a}\partial_{b}\partial_{c}\partial_{d}\partial_{e}\partial_{f}log\mathcal{L}\right)_{0}$
$\displaystyle-\left[\left\langle\partial_{a}\partial_{b}\partial_{c}h|\partial_{d}\partial_{e}\partial_{f}h\right\rangle+(19\
{\rm
perms})\right]-\left[\left\langle\partial_{a}\partial_{b}h|\partial_{c}\partial_{d}\partial_{e}\partial_{f}h\right\rangle+(14\
{\rm perms})\right]$
$\displaystyle-\left[\left\langle\partial_{a}h|\partial_{b}\partial_{c}\partial_{d}\partial_{e}\partial_{f}h\right\rangle+(5\
{\rm perms})\right]\,,$

contributing to the likelihood

$\displaystyle log\mathcal{L}_{6}$
$\displaystyle=-\frac{1}{6!}\sum_{a,\cdots,f}\mathcal{H}_{abcdef}\Delta\theta^{abcdef}$
$\displaystyle=-\sum_{a,\cdots,f}\left[\frac{1}{72}\left\langle\partial_{a}\partial_{b}\partial_{c}h|\partial_{d}\partial_{e}\partial_{f}h\right\rangle+\frac{1}{48}\left\langle\partial_{a}\partial_{b}h|\partial_{c}\partial_{d}\partial_{e}\partial_{f}h\right\rangle\right.$
$\displaystyle\qquad\qquad\left.+\frac{1}{120}\left\langle\partial_{a}h|\partial_{b}\partial_{c}\partial_{d}\partial_{e}\partial_{f}h\right\rangle\right]\Delta\theta^{abcdef}\,.$

Collecting the 3-derivative terms from Quarxion, P- and H-tensors we have the
Triplet contribution to the likelihood:

$\displaystyle\left.\log{\cal L}\right|_{Triplet}=$
$\displaystyle-\frac{1}{6}\sum_{a,b,c,d}\left\langle\partial_{a}h|\partial_{b}\partial_{c}\partial_{d}h\right\rangle\Delta\theta^{abcd}-\frac{1}{12}\sum_{a,b,c,d,e}\left\langle\partial_{a}\partial_{b}h|\partial_{c}\partial_{d}\partial_{e}h\right\rangle\Delta\theta^{abcde}$
(42)
$\displaystyle-\frac{1}{72}\sum_{a,\dots,f}\left\langle\partial_{a}\partial_{b}\partial_{c}h|\partial_{d}\partial_{e}\partial_{f}h\right\rangle\Delta\theta^{abcdef}$
(43)

##  Appendix B Example of parameter recovery

For the analysis of subec. 4.1 we report in Figs. 9,10,11, the relative bias,
i.e. the relative difference between the injected and maximum likelihood
estimator recovered with different likelihood derivative approximations and
with the exact likelihood, running the Nestle [35] MonteCarlo sampler on each
of them.

Figure 9: $d_{L}$ relative bias for the same injections as in Fig. 6.

Figure 10: Same as in Fig. 9, but for $z=0.5$ ($d_{L}\simeq 2.8$Gpc).

Figure 11: Same as in Fig. 9, but for $z=1$ ($d_{L}\simeq 6.6$Gpc).

## Acknowledgments

The authors thank Valerio Marra for useful discussions. JMSdS is supported by
the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) –
Graduate Research Fellowship/Code 001. The work of RS is partially supported
by CNPq under grant 310165/2021-0 and by FAPESP grants 2021/14335-0 and
2022/06350-2. The authors thank the High Performance Computing Center (NPAD)
at UFRN for providing computational resources that made the present work
possible.

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11institutetext: Haus der Astronomie, Campus MPIA, Königstuhl 17, D-69117
Heidelberg, Germany
11email<EMAIL_ADDRESS>
# Navigation in the Ancient Mediterranean and Beyond

M. Nielbock

(Received August 8, 2016; accepted March 2, 2017)

This lesson unit has been developed within the framework the EU Space
Awareness project. It provides an insight into the history and navigational
methods of the Bronze Age Mediterranean peoples. The students explore the link
between exciting history and astronomical knowledge. Besides an overview of
ancient seafaring in the Mediterranean, the students explore in two hands-on
activities early navigational skills using the stars and constellations and
their apparent nightly movement across the sky. In the course of the
activities, they become familiar with the stellar constellations and how they
are distributed across the northern and southern sky.

###### Key Words.:

navigation, astronomy, ancient history, Bronze Age, geography, stars, Polaris,
North Star, latitude, meridian, pole height, circumpolar, celestial
navigation, Mediterranean

##  1 Background information

###  1.1 Cardinal directions

The cardinal directions are defined by astronomical processes like diurnal and
annual apparent movements of the Sun and the apparent movements of the stars.
In ancient and prehistoric times, the sky certainly had a different
significance than today. This is reflected in the many myths all around the
world. As a result, we can assume that the processes in the sky have been
watched and monitored closely. In doing this, the underlying cycles and
visible phenomena were easy to observe.

Figure 1: Apparent diurnal movement of the Sun in the northern hemisphere at
equinox. The Sun reaches its highest elevation above the horizon to the south.
In the southern hemisphere, the Sun culminates to the north (Tau’olunga,
https://commons.wikimedia.org/wiki/File:Equinox-50.jpg, “Equinox-50”,
horizontal coordinate system and annotations added by Markus Nielbock,
https://creativecommons.org/licenses/by-sa/3.0/legalcode).

For any given position on Earth except the equatorial region, the Sun always
culminates towards the same direction (Fig. 1). The region between the two
tropics 23$\aas@@fstack{\circ}$5 north and south of the equator is special,
because the Sun can attain zenith positions at local noon throughout the year.
During night, the stars rotate around the celestial poles. Archaeological
evidence from prehistoric eras like burials and the orientation of buildings
demonstrates that the cardinal directions were common knowledge in a multitude
of cultures already many millennia ago. Therefore, it is obvious that they
were applied to early navigation. The magnetic compass was unknown in Europe
until the 13th century CE (Lane 1963).

###  1.2 Latitude and longitude

Any location on an area is defined by two coordinates. The surface of a sphere
is a curved area, but using coordinates like up and down does not make much
sense, because the surface of a sphere has neither a beginning nor an ending.
Instead, we can use spherical polar coordinates originating from the centre of
the sphere with the radius being fixed (Fig. 2). Two angular coordinates
remain. Applied to the Earth, they are called the latitude and the longitude.
Its rotation provides the symmetry axis. The North Pole is defined as the
point, where the theoretical axis of rotation meets the surface of the sphere
and the rotation is counter-clockwise when looking at the North Pole from
above. The opposite point is the South Pole. The equator is defined as the
great circle half way between the two poles.

Figure 2: Illustration of how the latitudes and longitudes of the Earth are
defined (Peter Mercator, djexplo, CC0).

The latitudes are circles parallel to the equator. They are counted from
$0\degr$ at the equator to $\pm 90\degr$ at the poles. The longitudes are
great circles connecting the two poles of the Earth. For a given position on
Earth, the longitude going through the zenith, the point directly above, is
called the meridian. This is the line the Sun apparently crosses at local
noon. The origin of this coordinate is defined as the Prime Meridian, and
passes Greenwich, where the Royal Observatory of England is located. From
there, longitudes are counted from $0\degr$ to $\pm 180\degr$.

Example: Heidelberg in Germany is located at 49$\aas@@fstack{\circ}$4 North
and 8$\aas@@fstack{\circ}$7 East.

###  1.3 Elevation if the pole (pole height)

If we project the terrestrial coordinate system of latitudes and longitudes at
the sky, we get the celestial equatorial coordinate system. The Earth’s
equator becomes the celestial equator and the geographic poles are
extrapolated to build the celestial poles. If we were to make a photograph
with a long exposure of the northern sky, we would see from the trails of the
stars that they all revolve about a common point, the northern celestial pole
(Fig. 3).

Figure 3: Trails of stars at the sky after an exposure time of approximately 2
hours (Ralph Arvesen, Live Oak star trails,
https://www.flickr.com/photos/rarvesen/9494908143,
https://creativecommons.org/licenses/by/2.0/legalcode).

Figure 4: Configuration of the two constellations Ursa Major (Great Bear) and
Ursa Minor (Little Bear) in the northern sky. Polaris, the North Star, which
is close to the true celestial north pole, is the brightest star in Ursa Minor
(Bonč, https://commons.wikimedia.org/wiki/File:Ursa_Major_-_Ursa_Minor_-
_Polaris.jpg, “Ursa Major – Ursa Minor – Polaris”, colours inverted by Markus
Nielbock, https://creativecommons.org/licenses/by-sa/3.0/legalcode).

In the northern hemisphere, there is a moderately bright star near the
celestial pole, the North Star or Polaris. It is the brightest star in the
constellation the Little Bear, Ursa Minor (Fig. 4). In our era, Polaris is
less than a degree off. However, 1000 years ago, it was $8\degr$ away from the
pole. Therefore, today we can use it as a proxy for the position of the
celestial North Pole. At the southern celestial pole, there is no such star
that can be observed with the naked eye. Other procedures have to be applied
to find it.

If we stood exactly at the geographic North Pole, Polaris would always be
directly overhead. We can say that its elevation would be (almost) $90\degr$.
This information already introduces the horizontal coordinate system (Fig. 5).
It is the natural reference we use every day. We, the observers, are the
origin of that coordinate system located on a flat plane whose edge is the
horizon. The sky is imagined as a hemisphere above. The angle between an
object in the sky and the horizon is the altitude or elevation. The direction
within the plane is given as an angle between $0\degr$ and $360\degr$, the
azimuth, which is usually counted clockwise from north. In navigation, this is
also called the bearing. The meridian is the line that connects North and
South at the horizon and passes the zenith.

Figure 5: Illustration of the horizontal coordinate system. The observer is
the origin of the coordinates assigned as azimuth and altitude or elevation
(TWCarlson, https://commons.wikimedia.org/wiki/File:Azimuth-
Altitude_schematic.svg, “Azimuth-Altitude schematic”,
https://creativecommons.org/licenses/by-sa/3.0/legalcode).

Figure 6: When combining the three coordinate systems (terrestrial spherical,
celestial equatorial, local horizontal), it becomes clear that the latitude of
the observer is exactly the elevation of the celestial pole, also known as the
pole height (own work).

Figure 7: Star charts of the northern celestial pole region for the years 2750
BCE and 2016 CE (own work, created with XEphem Version 3.7.6 produced by
Elwood C. Downey and distributed by the Clear Sky Institute Inc., Solon, Iowa,
USA, %****␣astroedu1645.tex␣Line␣200␣****http://www.xephem.com).

For any other position on Earth, the celestial pole or Polaris would appear at
an elevation smaller than $90\degr$. At the equator, it would just graze the
horizon, i.e. be at an elevation of $0\degr$. The correlation between the
latitude (North Pole = $90\degr$, Equator = $0\degr$) and the elevation of
Polaris is no coincidence. Figure 6 combines all three mentioned coordinate
systems. For a given observer at any latitude on Earth, the local horizontal
coordinate system touches the terrestrial spherical polar coordinate system at
a single tangent point. The sketch demonstrates that the elevation of the
celestial North Pole, also called the pole height, is exactly the northern
latitude of the observer on Earth. From this we can conclude that if we
measure the elevation of Polaris, we can determine our latitude on Earth with
reasonable precision.

###  1.4 Circumpolar stars and constellations

In ancient history, e.g. during the Bronze Age, Polaris could not be used to
determine north. Due to the precession of the Earth’s axis, it was about
$30\degr$ away from the celestial North Pole in 3,500 BCE. Instead, the star
Thuban ($\alpha$ Draconis) was more appropriate, as it was less than $4\degr$
off. However, it was considerably fainter than Polaris and perhaps not always
visible to the naked eye.

When looking at the night sky, some stars within a certain radius around the
celestial poles never set; they are circumpolar (see Fig. 3). Navigators were
skilled enough to determine the true position of the celestial pole, by
observing a few stars close to it. This method also works for the southern
celestial pole. There are two videos that demonstrate the phenomenon.

CircumpolarStars Heidelberg 49degN (Duration: 0:57)
https://youtu.be/uzeey9VPA48

CircumpolarStars Habana 23degN (Duration: 0:49)
https://youtu.be/zggfQC_d7UQ

They show the movement of the night sky when looking north for two different
latitudes coinciding with the cities of Heidelberg, Germany ($49\degr$ North)
and Lisbon, Portugal ($23\degr$ North). The videos illustrate that

  1. 1.

there are always stars and constellations that never set. Those are the
circumpolar stars and constellations.

  2. 2.

the angle between the celestial pole (Polaris) and the horizon depends on the
latitude of the observer. In fact, these angles are identical.

  3. 3.

the circumpolar region depends on the latitude of the observer. It is bigger
for locations closer to the pole.

If the students are familiar with the usage of a planisphere, they can study
the same phenomenon by watching the following two videos. They show the
rotation of the sky for the latitudes $20\degr$ and $4\degr$. The transparent
area reveals the visible sky for a given point in time. The dashed circle
indicates the region of circumpolar stars and constellations.

CircumPolarStars phi N20 (Duration: 0:37)
https://youtu.be/Uv-xcdqhV00

CircumPolarStars phi N45 (Duration: 0:37)
https://youtu.be/VZ6RmdzbpPw

When sailing north or south, sailors observe that with changing elevation of
the celestial pole the circumpolar range is altered, too. Therefore, whenever
navigators see the same star or constellation culminating – i.e. passing the
meridian – at the same elevation, they stay on the “latitude”. Although the
educated among the ancient Greek were familiar with the concept of latitude of
a spherical Earth, common sailors were probably not. For them, it was
sufficient to realise the connection between the elevation of stars and their
course. Ancient navigators knew the night sky very well. In particular, they
utilised the relative positions of constellation which helped them to
determine their position in terms of latitude.

###  1.5 Early seafaring and navigation in the Mediterranean

Navigation using celestial objects is a skill that emerged already long before
humans roamed the Earth. Today, we know numerous examples among animals who
find their course using the day or night sky. Bees and monarch butterflies
navigate by the Sun (Sauman et al. 2005), just like starlings do (Kramer
1952). Even more impressive is the ability of birds (Emlen 1970; Lockley 1967;
Sauer 1958) and seals (Mauck et al. 2008) to identify the position of stars
during night-time for steering a course. However, in our modern civilisation
with intense illumination of cities, strong lights can be mistaken for
celestial objects. For instance, moths use the moon to maintain a constant
course, but if confused by a street lamp, they keep on circling around it
until exhaustion (Stevenson 2008). Hence, light pollution is a serious threat
to many animals. Its magnitude is demonstrated by Fig. 8.

Figure 8: The Iberian Peninsula at night seen from the International Space
Station (Image courtesy of the Earth Science and Remote Sensing Unit, NASA
Johnson Space Center, mission-roll-frame no. ISS040-E-081320 (26 July 2014),
%****␣astroedu1645.tex␣Line␣300␣****http://eol.jsc.nasa.gov/SearchPhotos/photo.pl?mission=ISS040&roll=E&frame=081320).

Among the first humans to have navigated the open sea were the aboriginal
settlers of Australia some 50,000 years ago (Hiscock 2013). The oldest records
of seafaring in the Mediterranean date back to 7,000 BCE (Hertel 1990),
admittedly done with boats or small ships that were propelled by paddles only.
The routes were restricted close the coast where landmarks helped to navigate
to the desired destinations. In order to be able to cross larger distances,
propulsion independent of muscle force is needed. Therefore, the sail was one
of the most important inventions in human history, in its significance similar
to the wheel. Around the middle of the 4th millennium BCE, Egyptian ships
sailed the eastern Mediterranean (Bohn 2011) and established trade routes with
Byblos in Phoenicia, the biblical Canaan, now Lebanon. This is about the time
when the Bronze Age began. Tin is an important ingredient to bronze. After the
depletion of the local deposits, tin sources in central and western Europe
triggered large scale trade (Penhallurick 1986). Transportation over large
distances inside and outside the Mediterranean was accomplished by ships.

Figure 9: Map of the diffusion of metallurgy. Bronze Age tin deposits are
mostly found at the European Atlantic coast (User:Hamelin de Guettelet,
%****␣astroedu1645.tex␣Line␣325␣****https://commons.wikimedia.org/wiki/File:Metallurgical_diffusion.png,
public domain).

Soon, the navigators realised that celestial objects, especially stars, can be
used to keep the course of a ship. Such skills have been mentioned in early
literature like Homer’s Odyssey which is believed to date back to the 8th
century BCE. The original sources are thought to originate from the Bronze
Age, in which the Minoans of Crete were a particularly influential people that
lived between 3,650 and 1,450 BCE in the northern Mediterranean, and who
sailed the Aegean Sea. Since many of their sacral buildings were aligned with
the cardinal directions and astronomical phenomena like the rising Sun and the
equinoxes (Henriksson & Blomberg 2008, 2009), it is reasonable to think that
they used this knowledge for navigation, too (Blomberg & Henriksson 1999). The
Minoans sailed to the island of Thera and Egypt, which would have taken them
on open water for several days.

Figure 10: Map of Crete with ancient Minoan sites in the early 2nd millennium
BCE (Eric Gaba (Sting),
https://commons.wikimedia.org/wiki/File:Crete_integrated_map-en.svg,
https://creativecommons.org/licenses/by-sa/4.0/legalcode).

The Greek poet Aratos of Soli published his Phainomena around 275 BCE (Aratus
et al. 1921) in which he provided detailed positions of constellations and
their order of rising and setting, which would be vital information for any
navigator to maintain a given course. He would simply have pointed his ship at
a bearing and be able to keep it with the help of stellar constellations that
appeared towards that heading. The azimuth of a given star when rising or
setting remains constant throughout the year, except for a slow variation
caused by the 26,000 years period of the precession of the Earth’s axis.
Interestingly, Aratos’ positions did not fit the Late Bronze and Early Iron
Age but the era of the Minoan reign (Blomberg & Henriksson 1999) some 2,000
years earlier.

Figure 11: The night sky with bearing from Crete to Alexandria for 22
September 2000 BCE, 21:30 UT (own work, created with Stellarium, free GNU GPL
software, after Blomberg & Henriksson (1999), Fig. 9).

Around 1200 BCE, the Phoenicians became the dominating civilisation in the
Mediterranean. They built colonies along the southern and western coasts of
the Mediterranean and beyond. Among them was the colony of Gades (now Cadíz)
just outside the Strait of Gibraltar which served as a trading point for goods
and resources from Northern Europe (Cunliffe 2003; Hertel 1990). Several
documented voyages through the Atlantic Ocean took them to Britain and even
several hundred miles south along the African coast (Johnson & Nurminen 2009).

Figure 12: Trade routes of the Phoenicians during the European Bronze Age
(DooFi, https://commons.wikimedia.org/wiki/File:PhoenicianTrade_EN.svg,
https://creativecommons.org/licenses/by-sa/3.0/legalcode).

The Greek historian Herodotus (ca. 484 – 420 BCE) reports of a Phoenician
expedition funded by the Egyptian Pharaoh Necho II (610 -– 595 BCE) that set
out from the Red Sea to circumnavigate Africa and returned to Egypt via the
Mediterranean (Bohn 2011; Hertel 1990; Johnson & Nurminen 2009). The sailors
apparently reported that at times the Sun was located North (Cunliffe 2003)
which is expected after crossing the equator to the south. All this speaks in
favour of extraordinary navigational skills. After the Persians conquered the
Phoenician homeland in 539 BCE, their influence declined, but was re-
established by descendants of their colonies, the Carthaginians.

###  1.6 Pytheas

Figure 13: Statue of Pytheas, erected at the Palais de la Bourse in Marseille
in honour of his achievements (Rvalette,
https://commons.wikimedia.org/wiki/File:Pythéas.jpg, “Pythéas”,
https://creativecommons.org/licenses/by-sa/3.0/legalcode).

A very notable and well documented long distance voyage has been passed on by
ancient authors and scholars like Strabo, Pliny and Diodorus of Sicily. It is
the voyage of Pytheas (ca. 380 – 310 BCE), a Greek astronomer, geographer and
explorer from Marseille who around 320 BCE apparently left the Mediterranean,
travelled along the European west coast and made it up north until the British
Isles and beyond the Arctic Circle, during which he possibly reached Iceland
or the Faroe Islands that he called Thule (Baker & Baker 1997; Cunliffe 2003;
Hergt 1893).

Massalia (or Massilia), as it was called then, was founded by Phocean Greeks
around 600 BCE, and quickly evolved into one of the biggest and wealthiest
Greek outposts in the Western Mediterranean with strong trade relations to
Celtic tribes who occupied most of Europe (Cunliffe 2003). Pytheas was born
into the Late Bronze Age, when the trade with resources from Northern Europe
was flourishing. Not much was known in Greek geography about this part of the
world, except that the barbarians living there mined the tin ore and delivered
the precious amber that the whole Mediterranean so desperately longed for.
Perhaps it was out of pure curiosity why Pytheas set out to explore these
shores.

Figure 14: The journey of Pytheas of Massalia according to Cunliffe (2003)
(ESA/Cunliffe,
http://www.esa.int/spaceinimages/Images/2005/09/The_journey_of_Pytheas,
http://www.esa.int/spaceinimages/ESA_Multimedia/Copyright_Notice_Images).

His voyage was a milestone, because Pytheas was a scientist and a great
observer. He already used a gnomon or a sundial, which allowed him to
determine his latitude and measure the time during his voyage (Nansen 1911).
He also noticed that in summer the Sun shines longer at higher latitudes. In
addition, he was the first to notice a relation between the tides, which are
practically not present in the Mediterranean, and the lunar phases (Roller
2006).

##  2 List of material

The list contains items needed by one student. The teacher may decide that
they work in groups of two.

  * •

Worksheet

  * •

Pair of compasses

  * •

Pencil

  * •

Ruler

  * •

Calculator

  * •

Protractor

  * •

Torch (for supplemental third activity)

  * •

Magnetic compass (optional, third activity)

  * •

Computer with MS Excel installed

  * •

Excel spreadsheet:
astroedu1645_AncientMediterranean_BrightStars.xlsx

##  3 Goals

With this activity, the students will learn that

  * •

celestial navigation has been developed already many centuries ago.

  * •

apart from using Polaris there are other methods to determine cardinal
directions from the positions of stars.

  * •

ancient navigators were able to successfully navigate on open water following
stars and constellations.

##  4 Learning objectives

The students will be able to

  * •

describe methods to determine the cardinal directions form observing the sky.

  * •

name prominent stellar constellations.

  * •

explain the nature of circumpolar stars and constellations.

  * •

use an Excel spreadsheet for calculations.

  * •

describe the importance of improved navigational skills for early
civilisations.

##  5 Target group details

Suggested age range: 14 – 19 years
Suggested school level: Middle School, Secondary School
Duration: 90 minutes

##  6 Evaluation

According to the suggested questions listed in the description of the
activity, the teacher should guide the students to recognise the positions and
the apparent movement of celestial objects as indicators for cardinal
directions.

Before working on activity 1, the students should look closely at the map
provided. A visit at a planetarium helps remembering the constellations. Let
the students name constellations they already know.

Ask the students (see Q&A in the activity description) where the North Star
would be when observed from the terrestrial North Pole and the Equator. Then
ask them, how this position changes when travelling between these locations.
When that concept is understood, introduce the rotation and the apparent
motion of the stars. Show them the picture of the star trails and ask them,
where they come from. Ask them, which of the stars or constellations remain
above the horizon for the different locations on Earth mentioned above. Those
are circumpolar stars and constellations.

Explain the usage of the Excel spreadsheet needed for activity 2. Let the
students compare their results for different latitudes.

Discuss with the students, what the reasons for seafaring could have been in
ancient epochs.

The third optional activity acts as a wrap-up and can be used to evaluate,
what the students have understood.

##  7 Full description of the activity

###  7.1 Introduction

It would be beneficial, if the activity be included into a larger context of
seafaring, e.g. in geography, history, literature, etc.

Tip: This activity could be combined with other forms of acquiring knowledge
like giving oral presentations in history, literature or geography
highlighting navigation. This would prepare the field in a much more
interactive way than what a teacher can achieve by summarising the facts.

There are certainly good documentaries available on sea exploration that could
be shown. As an introduction to celestial navigation in general and the early
navigators, let the students watch the following videos. The last one is in
French. This could be done in junction with French lessons in school. If not,
tell the story about Pytheas as outlined in the background information. A link
to literature or history classes may be established by reading “The
Extraordinary Voyage of Pytheas” by B. Cunliffe.

Episode 2: Celestial Navigation (Duration: 4:39)
https://www.youtube.com/watch?v=DoOuSo9qElI

How did early Sailors navigate the Oceans? — The Curious Engineer (Duration:
6:20)
https://www.youtube.com/watch?v=4DlNhbkPiYY

World Explorers in 10 Minutes (Duration: 9:59)
https://www.youtube.com/watch?v=iUkOfzhvMMs

Once upon a time … man: The Explorers - The first navigators (Duration: 23:13)
https://www.youtube.com/watch?v=KuryXLnHsEY

Pythéas, un Massaliote méconnu (French, duration: 9:57)
https://www.youtube.com/watch?v=knBNHbbu-ao

Ask the students, if they had an idea for how long mankind already uses ships
to cross oceans. One may point out the spread of the Homo sapiens to islands
and isolated continents like Australia.

Possible answers:
We know for sure that ships have been used to cross large distances already
since 3,000 BCE or earlier. However, the early settlers of Australia must have
found a way to cross the Oceans around 50,000 BCE.

Ask them, what could have been the benefit to try to explore the seas.
Perhaps, someone knows historic cultures or peoples that were famous sailors.
The teacher can support this with a few examples of ancient seafaring peoples,
e.g. from the Mediterranean.

Possible answers:
Finding new resources and food, trade, spirit of exploration, curiosity.

Table 1: List of cities along with their latitudes. The solutions from activity 1 are added in italic writing. City | Latitude ($\degr$) | Radius in map (cm) | Constellations
---|---|---|---
Tunis (ancient | 36.8 | 3.7 | Ursa Minor, Ursa Major (Big Dipper), Draco, Cepheus, Cassiopeia
Carthage, Tunisia) |  |  |
Cape Town | -33.9 | 3.4 | Crux, Pavo (Peacock), Achernar, most of Carina, Toliman
(South Africa) |  |  | of Centaurus
Plymouth | 50.4 | 5.0 | Ursa Minor, Ursa Major, Draco, Cepheus, Cassiopeia, Perseus
(UK) |  |  | (mostly), part of Cygnus with Deneb
Wellington | -41.3 | 4.1 | Carina (incl. Canopus), Ara, Pictor, Dorado
(New Zealand) |  |  |
Mumbai | 19.0 | 1.9 | Ursa Minor
(India) |  |  |
Grytviken | -54.3 | 5.4 | Centaurus, Lupus, Main part of Puppis, Phoenix, Grus
(South Georgia) |  |  |

Ask the students, how they find the way to school every day. What supports
their orientation to not get lost? As soon as reference points (buildings,
traffic lights, bus stops, etc.) have been mentioned ask the students how
navigators were able to find their way on the seas. In early times, they used
sailing directions in connection to landmarks that can be recognised. But for
this, the ships would have to stay close to the coast. Lighthouses improved
the situation. Magnetic compasses have been a rather late invention around the
11th century CE, and they were not used in Europe before the 13th century. But
what could be used as reference points at open sea? Probably the students will
soon mention celestial objects like the Sun, the Moon and stars.

Suggested additional questions, especially after showing the introductory
videos:

Q: Who was Pytheas?
A: He was an ancient Greek scientist and explorer.

Q: Where and when did he live?
A: He lived in the 4th century BCE during the Late Bronze Age in Massalia, now
Marseille.

Q: Where did he travel?
A: Pytheas travelled north along the Atlantic coast of Europe to Britain and
probably to the Arctic Circle and Iceland.

Q: What did he observe and discover during his voyage?
A: He was the first Greek to travel so far to the north. He noticed that the
length of daylight depends on latitude. He was also the first to relate the
tides to the phases of the moon.

###  7.2 Activity 1: Circumpolar constellations and stars

In absence of a bright star at the celestial poles, ancient navigators were
able to find it by observing a few circumpolar stars. These navigators were
experienced enough to determine true north by recognising the relative
position of such stars and by their paths around it.

In addition, they used circumpolar constellations and stars to infer their
latitude. This means, they never rise or set – they are always above the
horizon. While today we can simply measure the elevation of Polaris above the
horizon, ancient navigators saw that star many degrees away from the celestial
North Pole. In the southern hemisphere, there is no such stellar indicator
anyway. So, instead of measuring the elevation of Polaris, they observed which
stars and constellations were still visible above the horizon when they
attained their lowest elevation above the horizon (lower culmination) during
their apparent orbit around the celestial pole.

Let the students watch the two following videos that demonstrate the
phenomenon of circumpolar stars and constellations for two locations on Earth.
They show the simulated daily apparent rotation of the sky around the northern
celestial pole.

CircumpolarStars Heidelberg 49degN (Duration: 0:57)
https://youtu.be/uzeey9VPA48

CircumpolarStars Habana 23degN (Duration: 0:49)
https://youtu.be/zggfQC_d7UQ

The students will notice that

  1. 1.

there are always stars and constellations that never set. Those are the
circumpolar stars and constellations.

  2. 2.

the angle between the celestial pole (Polaris) and the horizon depends on the
latitude of the observer. In fact, these angles are identical.

  3. 3.

the circumpolar region depends on the latitude of the observer. It is bigger
for locations closer to the pole.

If the students are familiar with the usage of a planisphere, they can study
the same phenomenon by watching the following two videos.

CircumPolarStars phi N20 (Duration: 0:37)
https://youtu.be/Uv-xcdqhV00

CircumPolarStars phi N45 (Duration: 0:37)
https://youtu.be/VZ6RmdzbpPw

They show the rotation of the sky for the latitudes $20\degr$ and $45\degr$.
The transparent area reveals the visible sky for a given point in time. The
dashed circle indicates the region of circumpolar stars and constellations.

Figure 15: Star charts of the northern and the southern hemisphere (Markus
Nielbock,
https://commons.wikimedia.org/wiki/File:NorthernCelestialHemisphere.png,
https://commons.wikimedia.org/wiki/File:SouthernCelestialHemisphere.png,
https://creativecommons.org/licenses/by/4.0/legalcode, created with PP3,
http://pp3.sourceforge.net)

####  7.2.1 Questions

Q: What is special about the geographic North and South Poles of the Earth
compared to other locations?
A: They define the rotation axis of the Earth.

Q: How do you find North and the other cardinal directions without a compass?
A: Celestial bodies, e.g. stars like Polaris which indicates the celestial
north pole.

Q: Why does the North Star (Polaris) indicate North?
A: In our lifetime, Polaris is close to the celestial north pole.

Q: Where in the sky would be the celestial North/South Pole if you stood
exactly at the terrestrial North/South Pole?
A: At the zenith, i.e. directly overhead.

Q: How would this position change, if you travelled towards the equator?
A: Its elevation would decline from zenith to the horizon.

Q: What are circumpolar constellations?
A: These are constellations that revolve around one of the celestial poles and
never rise or set. They are always above the horizon.

Q: Which of the visible constellations would be circumpolar, if you stood on
the terrestrial North/South Pole/equator?
A: The entire northern/southern hemisphere (poles). None at the equator.

Q: If the North Star was not visible, how would you be able to determine you
latitude anyway?
A: Since the circumpolar stars and constellations depend on the latitude, just
like the elevation of Polaris, the ones that always stay above the horizon
indicate, where I am.

####  7.2.2 Exercise

The task is now to walk in the footsteps of a navigator that lived around
5,000 years ago. Based on those skills, the students will determine the
constellations that are circumpolar when observed from given positions on
Earth.

The table below contains the names of six cities along with their latitudes
$\varphi$. Negative values indicate southern latitudes. A seventh row is
empty, where the students can add the details of their home town. From this,
they will have to calculate the angular radii $\varrho$ from the celestial
pole. The calculation is simple, because is the same as the pole height and
the latitude:

$\varphi=\varrho$

Then they select the map that matches the hemisphere. The students use the
compasses to draw circles of those radii around the corresponding pole. The
constellations inside that circle are circumpolar. The constellations that are
just fully or partially visible for a given city are added to the table.

Possible solutions are added in italics. The table prepared for the exercise
is contained in the worksheet.

####  7.2.3 Detailed instructions

  1. 1.

Determine the map scale. The angular scale is $90\degr$ from the poles to the
outer circle, i.e. the celestial equator.

  2. 2.

Convert the latitudes in the table into radii in the scale of the maps and add
them to the table.

  3. 3.

For each of the cities:

    1. (a)

Select the suitable map.

    2. (b)

Use the compasses to draw a circle with a radius that was determined for that
city.

    3. (c)

Find and note the visible circumpolar constellations. If they are too many,
just select the most prominent ones.

####  7.2.4 Discussion

In ancient times, Polaris did not coincide with the celestial North Pole.
Explain the importance of circumpolar stars and constellations for ancient
navigators.

Possible result:
They provided an excellent tool to maintain latitude and helped to not get
lost at open sea.

####  7.2.5 Solutions

The map scale is: $1\,{\rm cm}\propto 10\degr$

Figure 16: Solution for the northern and southern locations.

###  7.3 Activity 2: Stars guide the way

In the absence of a star like Polaris that indicates a celestial pole, ancient
navigators used other stars and constellations to determine cardinal
directions and their ship’s course. They realised that the positions where
they appear and disappear at the horizon (the bearings) do not change during a
lifetime. Experienced navigators knew the brightest stars and constellations
by heart.

####  7.3.1 Questions

Q: Can you determine the cardinal directions from other stars than Polaris?
Note that there is no star at the South Pole.
A: Yes. If you know the stars and constellations, they can guide the way, as
they return to the same positions each day.

Q: Why can you use rising and setting stars and constellations to steer a
course on sea?
A: The position at the horizon when rising and setting does not change (except
for a very slow long term variation).

Q: Would you be able to see the same stars every night during the year?
A: No, the time of rise and set changes. Stars visible during winter nights
are up during summer daytimes.

Figure 17: Bearings of selected rising bright stars for latitude $45\degr$ at
an elevation of $10\degr$ above the horizon (own work).

####  7.3.2 Exercise

The students will produce a stellar compass similar to Fig. 17. The
calculations that are needed to convert the sky coordinates of the stars into
horizontal coordinates, i.e. azimuth and elevation, are pretty complex.
Therefore, this activity comes with an Excel file that does it for them. It
consists of 57 bright stars plus the Pleiades which is a very prominent group
of stars.

All they have to do is entering the latitude of their location and the
elevation of the stars in the corresponding line at the bottom of the
spreadsheet. For the elevation, $10\degr$ is a good value. This means, they
will get the azimuths of the stars when observed at an elevation of $10\degr$.
One can also use different values, but this exercise is meant for finding
stars that just rise or set. The azimuth is an angle along the horizon,
counting clockwise from North.

Figure 18: Students’ template for creating their own stellar compass (own
work).

The last two columns (AZ1, AZ2) then display two azimuths, one when the star
is rising and one when the star is setting. Note that the distribution of
azimuths for rising and setting stars is symmetric relative to the meridian,
i.e. the line that connects North and South. The cells that show #NA do not
contain valid numbers. These stars never rise or set. They are either
circumpolar or below the horizon.

The students translate the values into the stellar compass below. They use a
protractor and indicate the position of each star on the circle. Then they
write its name next to it.

####  7.3.3 Discussion

One of the methods to navigate through the ancient Mediterranean was to stay
close to the shores. Besides the danger of shallow waters, explain why Bronze
Age mariners must have had methods that would have allowed them to safely
navigate on open waters. You may want to look at a map of the Mediterranean.

Possible answers: The ancient peoples also visited islands for trade or other
reasons. Many of them are not visible from coastlines of the Mediterranean.
The voyages often would also last longer than just a few hours. Vessels of
that age were able to pass five nautical miles per hour on average. There are
also reports that were passed on through the ages which tell us about
celestial navigation.

###  7.4 Activity 3: Do it yourself! (Supplemental)

Nothing is more instructive than actually applying to real conditions what has
been learned and exercised in theory. Therefore, the results from the previous
two activities can be tested in the field by observing the night sky.

This activity can be done by the students themselves at home or as a group
event with the class.

Select a clear evening and a site with a good view to the horizon. As soon as
it is dark enough to see the stars, let the students use their dimmed torches
to inspect their maps with the circumpolar ranges from activity 1. A dimmed
torch - even better: a red one – helps to keep the eyes adapted to the dark.

After identifying the brightest stars, let them use their stellar compasses
from activity 2. The students should point the markers of one or some of the
stars to the stars at the sky. Let them identify North (or South, depending on
which celestial pole is visible from your location). If in the northern
hemisphere, does this match the direction to the North Star, Polaris? In the
southern hemisphere, a magnetic compass might be needed.

Let the students identify the constellations they see in the sky on their
maps. Ask them to look North (South in the southern hemisphere) and name the
stars and constellations that are just above the horizon. Does this coincide
with the maps? Note that there should be a circle that indicates the
circumpolar range for the local latitude.

Try to highlight that by doing this activity, they are working like the
navigators from 4,000 years ago.

##  8 Connection to school curriculum

This activity is part of the Space Awareness category “Navigation Through The
Ages” and related to the curricula topics:

  * •

Coordinate systems

  * •

Basic concepts, latitude, longitude

  * •

Celestial navigation

  * •

Constellations

  * •

Instruments

##  9 Conclusion

The students learn about navigational methods and seafaring of the ancient
epochs like the Bronze Age. Within two activities (plus one optional
supplemental activity), they will learn how the apparent diurnal paths of
stars can help to find the cardinal directions and to set course to known
destinations in the Mediterranean.

###### Acknowledgements.

This resource was developed in the framework of Space Awareness. Space
Awareness is funded by the European Commission’s Horizon 2020 Programme under
grant agreement no. 638653.

## References

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## Glossary

##  Appendix A Supplemental material

This unit is part of a larger educational package called “Navigation Through
the Ages” that introduces several historical and modern techniques used for
navigation. An overview is provided via:

Navigation_through_the_Ages.pdf

The supplemental material is available online via the Space Awareness project
website at http://www.space-awareness.org. The direct download links are
listed as follows:

  * •

Worksheets: astroedu1645-Ancient-Mediterranean-WS.pdf

  * •

Excel file: astroedu1645_AncientMediterranean_BrightStars.xlsx