This article was inspired by the beautiful 2016 movie *Hidden Figures* (based on the book of the same name by M. L. Shetterley) which tell the dramatic story of three talented black women scientist that worked as “human computers” for NASA in 1961 for the Mercury project.

In the movie, the mathematician Katherine Goble (interpreted by Taraji P. Henson), had a brilliant intuition on how to numerically solve the complex problem to find the transfer trajectory for the reentry into the Earth atmosphere of the Friendship 7 capsule with the astronaut John Glenn on board. In the particular scene, she was standing together with other engineers and the director of the Langley Research Center (a fictional character interpreted by Kevin Coster) in front of the vast blackboard looking to graph and equations when she says that the solution might be in the “old math” and she runs to take an old book from a bookshelf with the description of the Euler method. The scene is also nicely described in the youtube video lesson by Prof. Alan Garfinkel of the UCLA. A detailed description of the numerical solution based on the original derivation of K. Globe is in the Wolfram blog website.

Katherine Globe was using for these complex calculation her brilliant brain with the support of a mechanical calculator (the Friden STW-10, in the movie, this machine is visible in different scenes). In a scene of the film, she revealed that her typical computing performance was of 10000 calculations per day and probably for calculations, she was not referring to single arithmetic operations! These exceptional mathematical skills have given a significative contribution at the beginning of the American space program, but it became insufficient to handle the more complex mathematics necessary to land the man on the Moon, and the other fantastic NASA achievements.

Therefore, at the end of the movie, it is shown that the Langley Research Center to keep pace with the progress of the technology the acquire one of just developed electronic computer: the IBM 7090. This computer was one of IBM’s first transistor-based computers specifically designed for the scientific computing that NASA needed for the Mercury and Gemini space missions. The processing speed of this computer was in the range of 100-200 Kflops/s with costs of millions of dollars, and definitively not easily transportable! Today, the smartphone in our pocket is easily tens of thousand times more powerful and less expensive!**Euler’s Method**

Going back to the Euler method, let now see how it can be used to calculate trajectories of satellites by numerically integrating the law of gravity.

Leonhard Euler (1707-1783) was a genius polymath Swiss scientist and one of the most prolific mathematicians in history. To date, his complete opera, accounting thousand of papers written in Latin is yet not completely translated. The method to calculate approximately the solution of ordinary differential equations was reported in the book *Institutionum calculi integralis* (** Foundations of Integral Calculus**) published in 1768.

An ordinary differential equation (ODE) is an algebraic expression that put in relation the derivative of a dependent variable, with respect an independent one, to a function of both variables. For example in the differential equation

with the dependent variable and , the independent one. The equation tells us that there is a function of such that the first derivative is equal to Being an equation, it has solutions. However, the solutions are not numbers but functions.

Most problems in physics and engineering appear in the form of ordinary differential equations. For example, the motion of a particle is described by Newton’s equations, which is a second-order ordinary differential equation involving at least a second-order derivative in time, and the motion of a quantum particle is described by the Schrodinger equation, which is a partial differential equation involving a first-order partial derivative in time and second-order partial derivatives in coordinates.

The Initial-Value problem

An initial-value problem consists in the solution of (systems) DE’s describing dynamical systems as, for example, the motion of the Moon, Earth, and the Sun, the dynamics of a rocket, or the propagation of ocean waves, given the initial position and velocities of the system components. Many DE’s can be solved using integration methods of Calculus. However, it is usually hard if not impossible, to find an analytical solution. Euler was well aware of this difficulty and gave the most straightforward method for approximating a solution that brings his name.

A straightforward system is a particle of mass m moving in one dimension along the x-axis under an elastic force *f(x)*. In this case, the Newton’s equation that governs the dynamics of the system is given by

where *a* and *v* are the acceleration and velocity of the particle, respectively, and *t* is the time. If we divide the time into small, equal intervals h = t_{i +1}− t_{i}, elementary physics tell us that the velocity at time is approximately given by the average velocity in the time interval

the corresponding acceleration is approximately given by the average acceleration in the same time interval,

as long as τ is small enough. The simplest algorithm for finding the position and velocity of the particle at time t_{i+1}from the corresponding quantities at time ti is obtained after combining the previous equations, and we have

where If the initial position and velocity of the spring-bound particle is given (the initial-value problem), we can recursively calculate the particles trajectories from the algorithm given in Eq. (2). This algorithm is commonly known as the **Euler method** for the initial-value problem. This simple example illustrates how most algorithms are constructed. First, physical equations are transformed into discrete forms, namely, difference equations. Then the desired physical quantities or solutions of the equations at different variable points are given recursively with the quantities at a later point expressed in terms of the quantities from earlier points. In the above example, the position and velocity of the particle at are given by the position and velocity at , provided that the force at any position is explicitly given by a function of the position.

Note that the above way of constructing an algorithm is not limited to one-dimensional or single-particle problems. In fact, we can immediately generalize this algorithm to two-dimensional and three-dimensional problems, or to the problems involving more than one particle, such as the motion of a projectile or a system of three charged particles. The generalized version of the above algorithm is

where **R = (r _{1}**

**,r**

_{2}**,…,r**

_{n}**)**is the position vector of all the n particles in the system;

**V = (v**

_{1}**,v**

_{2}**,…,v**

_{n}**)\**and

**A = (a**

_{1}**,a**

_{2}**,…,a**

_{n}**)**, with

**a**

_{j}**= f**/m

_{j}_{j}for j = 1, 2, . . . , n, are the corresponding velocity and acceleration vectors, respectively. I will come back in another post about the numerical solution of DE systems.

The Euler’s method is based on the Taylor series representation of the DE solution. You can find more about Taylor series in another post, here we just recall the Taylor theorem

I*f and its derivatives for are continuous on the closed interval and is differentiable on the open interval then there exists such that *

The first sum on the right side of the equation is called *the Taylor Polynomial* *of degree n* and may be used to approximate . The last term is called *the reminder term* and it can be used to estimate the total error of the polynomial approximation of the function

The Euler method is based on the first order Taylor polynomial approximation () of the solution on the interval such that The Taylor expansion for *n=2* in is equal to

as from (2), by discarting the reminder, the expression (4) can be approximated to

with in compact notation. It is possible to show that the difference between the true value of the solution and the approximate is proportional to the increment *h*. In addition, the error affect the stability of the solution by increasing the deviation from the true value according to an amplification factor g (see [1]), i.e., if is the error at the step i, the error at the step is given by . In general, a numerical method is stable if a small error at any stage produces a smaller cumulative error [1].

Therefore it is possible in principle to increase the accuracy of the numerical solution by descreasing the value of the increment *h* but paying the price of increase the number of steps required to reach the maximum value of *t*.

We can use a simple program in awk language to study the effect of the step size, *h, *on the accuracy of the solution of the ODE in (1). Given the initial conditions , the solution of (1) is simple . The following awk script calculate the value of the solution at using 6 different time steps and compare the results with the analytical solution one calculate as

```
#======================================================================
#
# NAME: Euler.awk
#
#======================================================================
# DESCRIPTION: Test the effect of the time step on the numerical
# integration of the DE y'=cos(t) using the Euler method
#
#======================================================================
# DEVELOPED USING: gawk
# REQU. FUNCTIONS:
#======================================================================
# Author: Danilo Roccatano
# Version: 1.0
# Copyright (C): 2019 Danilo Roccatano
#======================================================================
#
function f(t) {
return cos(t)
}
BEGIN {
# Set the time steps to test
h[0]= 0.005
h[1]= 0.01
h[2]= 0.05
h[3]= 0.075
h[4]= 0.1
h[5]= 0.5
h[6]= 1.0
# Set the end value of the integration interval in (radiants)
T=1
# calculate the analytical solution a T
cval=sin(T)
printf "%8.3f \n",sin(1)
for (j=0;j<7;j++) {
hh=h[j]
t=0
N=T/hh
y=0
for (i=0;i<N;i++) {
t=t+hh
y=y+f(t)*hh
}
diff = cval-y
printf "%d %8.3f %3d %8.3f (%8.3f) \n",j+1,hh,N,y,diff
}
printf "\n"
}
```

The results are listed in the following table. By increasing the time step the solution become more unstable giving larger and larger deviations from the correct value. The method used in the script is called ** forward Euler method** and it belong to a class of method called

**.**

*explicit integration methods*# | TIme step | N. steps | Value | |Difference|* |

1 | 0.005 | 200 | 0.840 | 0.001 |

2 | 0.01 | 100 | 0.839 | 0.002 |

3 | 0.05 | 20 | 0.830 | 0.012 |

4 | 0.1 | 10 | 0.818 | 0.024 |

5 | 0.5 | 2 | 0.709 | 0.133 |

6 | 1.0 | 1 | 0.540 | 0.301 |

Using the Taylor expansion formalism, it is possible to introduce a simple modifications to the Euler method that substantially improve its accuracy. The simplest and most common of these algorithms is the **Verlet scheme**. To derive this scheme, we first consider the Taylor expansions at of the coordinates for a system of particles with mass m_{i}:

by adding the second to the first, it results:

Where, the second derivative of the position has been substituted by the forces *F** _{i}(t)* acting on the particle

*i*at the time t divided by their masses. Since the higher order terms have been neglect, this integration scheme can provide an accuracy of the fourth order in Δ

*t*. The velocities can be obtained as:

The error on the velocities is in the order of (Δ*t*)^{3}.

A variation of the Verlet scheme, commonly used to integrate the Newton equation of motion in dynamic systems, is the so-called *leap-frog *algorithm. This scheme calculate the coordinates every time step and the velocities at half time step.

In the Figure I a schematic representation of the algorithm is reported.

This method is much efficient from the computational point of view. It is simpler and requires less computational and memory resources concerning other methods.

Now, it is the time to come back to the main motivation of this Post (the Hidden Figures movie) and to use the tools that we have learned for solving a celestial mechanics problem. Namely, we are going to use the numerical integration to solve Newton’s equation for the motion of planets in our solar system.

In this case, the mathematical problem to solve is expressed by the following system of DE:

with and positions and velocities of the particles, respectivelly.

Also in this case, we will use the leap-frog method as a convenient approach to this type of problem [1]. In Appendix I, you find an implementation of the leap-frog algorithm in the awk language program called iPlanetario.awk. The program read the file containing the masses of the planets, their initial positions and velocities. As an example of the input file for the program with the starting position and velocities of the Sun and the nine planets (comprising Pluto) of the Solar System at A.D. 2019-Jun-27 00:00:00.0000 TDB. The positions (ephemeris) and speeds of planets in the file were obtained from Horizons online “Ephemeris System” of the Solar System Dynamics Group at the Jet Propulsion Laboratory in Pasadena (USA). To run the program, use the command:

*gawk -f iPlanetario.awk <solar_system.dat> <siml> <dt> <sf>,*

with *solar_system.dat* the data file reported in Appendix II, siml the length of the simulation in days, dt the integration time step in seconds and sf the frequency of writing trajectory data (coordinates and velocities) on the output data. The program read from the file the masses, intial positions and velocities of the planets (function Read_Input_File()). It then calculates the velocities at dt/2 using the method proposed in the volume I chapter 9 of the Feynman’s lectures [2] by calling the function Calc_starting_vel(). The integration of the equation of motion using the leap-frog method is performed for several steps equal to *siml*24*3600/dt*in the main part of the program. The trajectories of the single planets are generated as separated files named as the corresponding planet name (which is in the last column in the input file). The integration is done in time step of seconds; in the program, the variable is set to dt=24**3600*seconds.

*How well does this simple program produce the results?*

Shall we verify the short term accuracy by comparing the generated trajectories with those obtained from the JPL propulsory laboratory? The HORIZONS ephemeris JPL service [3] produce very accurate ephemeris of know solar bodies that includes 780,000+ asteroids, 3525 comets, 178 natural satellites, all planets, the Sun, 150 spacecraft, and several dynamical points. It allows calculating the trajectories of all the known objects in the solar systems. You need to use in the current settings the keyword **VECTORS **for the Ephemeris Type entry, as also shown in Figure 4.

The Horizons server generates a file containing the trajectory in steps of one day for each selected object. The coordinates have a solar system barycentric origin and use as a coordinate system the ecliptic and mean equinox of reference epoch complying with the ICRF/J2000.0 reference frame.

Using the first-day position and velocities, you can then prepare the input file for the *iPlanetario.awk *program like the one reported in Appendix II. In this case, the location and the speeds of the planets are the one on July 1st, 1969, 16 days before the launch of Apollo 11 mission to the moon. Shall now run the program till the end of July 1969 (31 days) using an integration step of dt=1h, we can compare the percentage difference with the JPL trajectory. Figure 5 and Figure 6 show the results of the comparison.

The first three columns show the average relative difference concerning the JPL trajectories, the last three the average absolute difference in Km. Comparing Figure 5 and 6, it is evident how decreasing the time step the accuracy increase. It is also apparent that accuracy depends on the mass of the planet and the velocities component. In this sense, giant planets are less perturbed in their orbits by the other bodies. Mercury is the most perturbed one.

This approximated simulation may have missed the Earth position for the Apollo 11 astronauts for only some hundreds of kilometres. Probably it is not accurate enough for the mission. Still, it shows that the humble Euler’s method can give significative results as the genial intuition of Katherine Globe did anticipate.

**BIBLIOGRAPHY**

- D. Potter. Computational Physics. Wiley-Blackwell, 1973.
- Feynman, Leighton, Sands. Feynman lectures on physics. http://www.feynmanlectures.caltech.edu. Chapter 9.
- Horizons online Ephemeris System. https://ssd.jpl.nasa.gov.

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**APPENDIX I**

Program in awk language to integrate the Newton equation of motion for celestial n-bodies system.

```
#================================================================================
#
# NAME: iPlanetario.awk
#
#================================================================================
# DESCRIPTION:
# This program performed a n-body simulation of a cluster of
# stars or planets. It use the leap-frog algorithm to integrate
# the Newton equation of motion.
#================================================================================
# I/O FILES :
# INPUT FILE: Read a std input file with the following format
# LINE 1: Title
# LINE 2: number of bodies (NB)
# LINES 3-NB : space separated following information
# <bodies masses> <X> <Y> <Z> <Vx> <Vy> <Vz> <Name>
# - masses are divided by 10^24 Kg
# - Position in astronomical units (AU)
# - Velocities in AU/Day
# Example:
# POSITION AND ORBITAL VELOCITIES OF SOLAR SYSTEM PLANETS
# 2
# 1 1.989e6 0. 0. 0. 0. 0. 0. SOLE
# 2 0.330 -3.016E-01 -3.311E-01 -2.298E-04 1.536E-02 -1.734E-02 -2.826E-03 MERCURY
#================================================================================
# OUTPUT FILE: The name of each body is used as filename containing the trajectory of the
# body.
# FILE EXAMPLE: P_MERCURY.out
# -3.00849259e-01 -3.31952335e-01 -3.71157034e-04
# -3.00077609e-01 -3.32815098e-01 -5.12462809e-04
# -2.99303492e-01 -3.33675124e-01 -6.53764370e-04
# ...
#======================================================================
# USAGE:
# gawk -f iPlanetario.awk <fname> <siml> <dt> <sf>
# <fname>: Name of the input file
# <siml> : Lenght of the simulation in days (default = 365)
# <dt> : time step in seconds (default dt=24*3600)
# <sf> : Frequency of saving trajectory data (coordinated and velocities)
# in seconds (default dt=24*3600)
#======================================================================
# Version: 1.0
# Created: 2019-07-04 20:53
# Copyright (C): 2019 Danilo Roccatano
#======================================================================
#======================================================================
#
# FUNCTIONS
#
function Calc_starting_vel() {
#
# Calculate the starting velocity at dt/2
#
for (i=0;i<Nb;i++) {
AX[i]=0.
AY[i]=0.
AZ[i]=0.
#
# Calculate all the forces on each planet
#
for (j=0;j<Nb;j++) {
if (j!=i) {
D2=(X[i]-X[j])^2+(Y[i]-Y[j])^2+(Z[i]-Z[j])^2
D=sqrt(D2)
Ac=-G*mass[j]/D2
AX[i]=AX[i]+Ac*(X[i]-X[j])/D
AY[i]=AY[i]+Ac*(Y[i]-Y[j])/D
AZ[i]=AZ[i]+Ac*(Z[i]-Z[j])/D
}
}
VX[i]=VX[i]+AX[i]*dt*0.5
VY[i]=VY[i]+AY[i]*dt*0.5
VZ[i]=VZ[i]+AZ[i]*dt*0.5
}
}
function Read_Input_File() {
#
# Read input file
#
getline < fname
getline < fname
Nb=$1
for (i=0;i<Nb;i++) {
getline < fname
mass[i]=$2*MU
X[i]=$3*AU
Y[i]=$4*AU
Z[i]=$5*AU
VX[i]=$6*VCV
VY[i]=$7*VCV
VZ[i]=$8*VCV
# read the nameof the system
sname[i]=$9
}
}
#==============================================================================
# MAIN
#==============================================================================
BEGIN {
#==============================================================================
# Constants
#
# AU : Astronimical Unit
# G : Universtal gravitation Constant
# MU : Mass unit
# K2M : Km to meter
# D2S : Days to second
# VCV : Convert velocities
# nsteps: total number of simulation step
##
G=6.67430e-11
AU=1.496e11 #m
MU=1e24
K2M=1000
D2S = 24*3600
VCV=AU/D2S
#################################################
# Read the command line
#################################################
#
# Name of the input file
fname = ARGV[1]
# Lenght of the simulation in days
ARGV[2]?siml=ARGV[2]:siml=365
# Time step in seconds
ARGV[3]?dt=ARGV[3]:dt=D2S
# Frequency of saving coordinates and velocities
ARGV[4]?sf=ARGV[4]:sf=D2S
nsteps=siml*D2S/dt
#################################################
# Read input file
#################################################
Read_Input_File()
#################################################
# WRITE THE SUMMARY OF SIMULATION PARAMETERS
#################################################
for (i=0;i<Nb;i++) {
print "#DATA FILE NAME : ", fname > "P_"sname[i]".out"
print "#SIMULATION LENGTH (days): ", siml > "P_"sname[i]".out"
print "#TIME STEP (s) : ", dt > "P_"sname[i]".out"
print "#NUMBER OF STEPS : ", nsteps > "P_"sname[i]".out"
print "#SAVING FREQUENCY (s) : ", sf > "P_"sname[i]".out"
}
##################################################
# Initialize the velocities at dt/2
##################################################
Calc_starting_vel()
##################################################
# Main Loop
##################################################
tt=0.0
for (t=1;t<=nsteps;t++) {
for (i=0;i<Nb;i++) {
# update the new position
X[i]=X[i]+VX[i]*dt
Y[i]=Y[i]+VY[i]*dt
Z[i]=Z[i]+VZ[i]*dt
# clean the accelerations
AX[i]=0
AY[i]=0
AZ[i]=0
#
# Calculate all the forces on each body
#
for (j=0;j<Nb;j++) {
if (j!=i) {
D2=(X[i]-X[j])^2+(Y[i]-Y[j])^2+(Z[i]-Z[j])^2
D=sqrt(D2)
Ac=-G*mass[j]/D2
AX[i]=AX[i]+Ac*(X[i]-X[j])/D
AY[i]=AY[i]+Ac*(Y[i]-Y[j])/D
AZ[i]=AZ[i]+Ac*(Z[i]-Z[j])/D
}
}
# update the new positions
VX[i]=VX[i]+AX[i]*dt
VY[i]=VY[i]+AY[i]*dt
VZ[i]=VZ[i]+AZ[i]*dt
if (tt%sf == 0) {
printf "%15.8e %15.8e %15.8e \n", X[i]/AU,Y[i]/AU,Z[i]/AU > "P_"sname[i]".out"
}
}
tt+=dt
}
}
```

**APPENDIX II**

**EXAMPLE OF INPUT FILE: **The starting position and velocities 9 planets of the Solar System at A.D. 1969-Jul-01 00:00:00.0000 TDB. The ephemeris have been obtained from Horizons online Ephemeris System of the Solar System Dynamics Group at the Jet Propulsion Laboratory in Pasadena (USA).

POSITION AND MEAN ORBITAL VELOCITIES OF SOLAR SYSTEM PLANETS (on 1/7/1969) 11 1 1.989e6 4.501422142050814E-03 8.265096431911833E-04 -6.106498754821928E-05 -3. 725998016167921E-07 5.629018018191787E-06 -1.420343330057636E-08 SOL 2 0.330 3.626352905971110E-01 -3.112613637670733E-02 -3.556949387212367E-02 -2.936459346589251E-03 2.929519404193396E-02 2.661199336367969E-03 MERCURY 3 4.872 6.434726285492286E-01 -3.454394151201869E-01 -4.166240919470099E-02 9.516484358480345E-03 1.770408177673933E-02 -3.091459518364233E-04 VENUS 4 5.972 1.707701318320183E-01 -1.002143528594855E+00 -1.256995866771295E-04 1.668497591651540E-02 2.752898669289915E-03 -8.323100509758512E-08 EARTH 5 0.642 -6.669858220909308E-02 -1.458105888577555E+00 -2.885595380269896E-02 1.450798593015611E-02 5.244375487322264E-04 -3.467167007807005E-04 MARS 6 1898 -5.376385639107029E+00 -8.726835067297002E-01 1.240832480992071E-01 1.122255348949764E-03 -7.096896640081010E-03 4.095137444408094E-06 JUPITER 7 568 7.884725882151399E+00 4.851428514519507E+00 -3.981641564307521E-01 -3.226967396201162E-03 4.740175258901204E-03 4.511263076949473E-05 SATURN 8 86.8 -1.826473469434352E+01 -1.177885710585230E+00 2.326769071350472E-01 2.237727077038867E-04 -4.108566058385769E-03 -1.826199809165856E-05 URANUS 9 102.0 -1.604710777852071E+01 -2.571063138942750E+01 8.990992317848128E-01 2.642815824514614E-03 -1.643946715455203E-03 -2.678926484808933E-05 NEPTUNE 10 0.0146 -3.048235507172484E+01 2.734455614187439E+00 8.524022429973490E+00 3.221949947061528E-04 -3.306300878753190E-03 2.734922740068579E-04 PLUTO 11 0.073 1.718200895379193E-01 -1.004286292375689E+00 -3.046288847913102E-04 1.726098619408255E-02 3.018131142325193E-03 2.792664511938875E-05 Moon