Acceleration inside an aircraft in parabolic flight

This paper aims to present and analyse the acceleration data inside an aircraft during a parabolic flight. The data used were obtained during the flight from the automatic recordings of the aircraft and from a portable data logger with a built-in 3-axis accelerometer connected to a graphic calculator. There is good agreement between the accelerations obtained by the two methods. Based on the altitude data collected during each parabolic manoeuvre performed by the aircraft, it was possible to estimate the gravity of Mars and the Moon, as well as the values of the acceleration of gravity during the moments of microgravity. The analysis presented can also help to improve the understanding of the concepts of inertial forces and the equivalence between gravity and acceleration.


Introduction
Parabolic flights are used to conduct scientific experiments and test space equipment technology in microgravity conditions.These type of flights are the only manned research platform capable of inducing weightlessness without leaving the Earth's atmosphere.For example, Plester [1] emphasises the importance of careful aircraft selection for scientific purposes while pointing out the unique and versatile nature of parabolic flights for gravityrelated research.The importance and increasing demand for this type of flights is clearly illustrated by the work of Ulltrich et al [2], who present a description of the parabolic flight programme in Switzerland, based on the Airbus A310 ZERO-G, and identifies some of the experimentscarried out aboard the Airbus A310 ZERO-G during the Swiss parabolic flight campaigns.Carr et al [3] highlight the importance of using parabolic flights to simulate the space environment and understanding its limitations.The same authors also highlight the commercial hardware solution for data acquisition, the availability of raw and calibrated data, and the methodology for characterizing g levels and corresponding durations achieved for 20 parabolas.
In an educational context, there are a number of programmes where parabolic flights have been used for the study of the effects of microgravity.For example, Pletser et al [4] present a distinctive approach involving secondary school teachers who facilitated simple experiments to teach students about microgravity.Callens et al [5] and Perez-Poch et al [6] both positively emphasise the use of this type of flight for student experiments purposes, [5] focusing especially on scientific experiments and [6] on biomedical research.Perez-Poch and González [7] present an innovative method for performing parabolic flights with a single-engine aerobatic aircraft, therefore enabling a range of research and educational experiments.Pletser [8] provides an historical perspective on Professor J C Legros involvement with microgravity research during parabolic flights, highlighting the diverse areas of fluid physics investigated.Serfontein et al [9] present a more recent case where space engineering students tested a drag sail concept in microgravity, with significant educational benefits.
Taken together, these studies not only confirm, but highlight the importance of parabolic flights in enriching student learning through microgravity research.
This article examines the acceleration data collected during a parabolic flight in Beja, Portugal, inside an Airbus A310.The experiments for the flight were planned and selected by a science club based at a secondary school in the North of Portugal.The material used was limited to what was available in the science club and approved by the flight crew.The selected material also allows students to analyse data collected in the classroom.
The parabolic flight was carried out within the framework of the 'Zero-G Portugal-Astronaut for a day' campaign promoted by the Portuguese Space Agency [10].In addition to the flight crew, passengers included 31 students aged between 14 and 17, ESA Astronaut Matthias Maurer and a secondary school physics teacher, among others (see figure 1).

The parabolic flight
During a parabolic flight, it is possible to achieve near-weightlessness (micro-g), lunar gravity (1.62 m s −2 ) and Martian gravity (3.71 m s −2 ) inside the aircraft for approximately 22 s, depending on the parabolas.Nevertheless, flight manoeuvres that simulate lunar and Martian gravity levels are usually achieved by approximating parabolic trajectories, but are still referred to (albeit inaccurately) as lunar and Martian parabolas.It is still acceptable to refer to these manoeuvres as parabolic because the aircraft ballistic free-fall trajectory approximated as an arc of an elliptical orbit in a central gravity field has a relative error of the order of 10 −4 , compared to the trajectory approximated as an arc of a parabola in a parallel gravity field [12,13].In addition to the effect of the gravitational field, air resistance is not taken into account [14,15] and may not be fully compensated by the aircraft's manoeuvres.
During the lunar and Martian parabolas, the engine thrust is reduced to a level where the remaining vertical acceleration inside the cabin is ∼0.16g (g = 9.80 m s −2 ) for about 25 s and ∼0.38g for about 32 s, while the injection angles are about 42 and 38 degrees, respectively [16].
Space experiments are often conducted on board parabolic flights to improve their quality and success rate.These flights also provide an opportunity to carry out scientific experiments in microgravity or reduced gravity conditions without the need for a more expensive space flight.In recent years, parabolic flights have become more accessible to the general public, therefore providing an opportunity to experience weightlessness.
This study focuses on the different gravitational environments (Mars, Moon and microgravity) created during the parabolic flight of an Airbus A310-304 operated by Novespace.
To execute the parabolic manoeuvre the aircraft must first attain an altitude greater than 6000 metres and a velocity of 820 km h −1 .Once the flight level is stabilised at 6000 metres, the aircraft increases its altitude.The parabolic manoeuvre can then be divided into three phases: the parabola pull-up, the parabola, and the parabola pull-out [11,17].
To achieve the microgravity parabola, during the pull-up phase, the pilots raise the aircraft's nose from horizontal flight to an angle of 45-50 degrees.This causes hypergravity for approximately 20 s, where passengers experience 1.8 to 2 times their weight on Earth's surface.Afterwards, the aircraft transitions to the parabola phase through an 'injection' manoeuvre.This means the aircraft adopts a parabolic-like trajectory with reduced thrust power, resulting in a decrease in engine velocity.Within 5 s, the vertical load factor decreases from 1.8g to approximately 0g.During the parabola phase, the pilots follow a specific procedure: the first pilot adjusts the control stick tilt to maintain the vertical load factor at zero (nose angle); the second pilot maintains the roll angle at zero, keeping the wings horizontal; the mechanic adjusts the thrust of the engines to cancel the thrust longitudinal load factor, compensating for the effect of air drag.The mechanic also monitors the flight parameters, including warnings, temperatures, and pressure [16,17].
The aircraft follows a ballistic trajectory, and weightlessness begins when it is in free fall for approximately 20-25 s during the parabola phase.The pilots maintain a near-zero acceleration level in all three axes to ensure zero-gravity precision of ±0.02g [17].
Finally, the pull-out phase, which is the descending part of the parabolic trajectory, is symmetrical to the pull-up phase.When entering the pull-out phase, the nose of the aircraft is tilted downward 42 degrees.The transition from the parabolic phase to the pull-out phase also takes approximately 5 s, as shown in figure 2. The pilots gradually level the aircraft as engine speed increases, and, once again, the passengers experience a pull of 1.8g for about 20 s until returning to a stabilized altitude level (see figure 2).
The cycle shown in figure 2 is repeated in accordance with the flight plan.
To ensure the safety and comfort of all passengers and crew, there is a two-minute gap between successive manoeuvres to allow for preparation and recovery.

Experimental set-up
Two identical experiments were carried out on board of the aircraft to collect data on the acceleration and the weight of the masses suspended (see figure 3).For each experiment, we recorded the acceleration and weight of suspended masses at one-second intervals during the parabolic manoeuvres.Data were collected using a portable data logger (CLAB) that was equipped with a 3-axis accelerometer and connected to a force sensor.The CLAB was connected to a CASIO graphing calculator (fx-CG50) for data recording.
The CLAB [18] is an interface that can be used for data logging with CASIO graphical calculators or computers.The interface selection was based on its size (97 mm × 108 mm × 26 mm), mass (213 g), incorporated battery, USB interface and integrated 3-axis  accelerometer (which measures in the ranges ±20 m s −2 , ±40 m s −2 and ±80 m s −2 ), with frequencies up to 400 Hz.When assembling the CLAB, it is important to avoid influencing the response of the accelerometer.For this reason, flight crew used double-sided adhesive tape to set it up, since it is practical and has low bias, given the low mass of the device.The acquisition was started and stopped manually using the control panel on the device.
The data collected was compared to the acceleration values obtained from the aircraft's automatic data collection system.

Results and discussion
This section presents the results of aircraft altitude and acceleration.The data were obtained automatically during flight by the aircraft's acquisition system and through the experiment set up described in the previous section (CLAB accelerometer).
Aircraft's automatic data  Figure 6 compares the acceleration of the parabolas obtained from the aircraft's automatic data (black line) with the data obtained from CLAB accelerometer (green line) for approximately 1000 s, including the first sequence and one 0g parabola.It is possible to see in figure 6 that the data compared are in agreement.The most significant deviations occur during the highest accelerations, especially during the pull-up and pull-out phases.For the pull-up and pull-out phases represented in the figure, the CLAB recorded an acceleration of 17.86 ± 0.54 m s −2 , while the aircraft acquisition system indicated 17.29 ± 0.51 m s −2 .The differences observed may be due to the fact that the acquisition systems had distinct placements: one was on the aircraft itself (acquisition system) and the other was part of the experiment mounted on the aircraft (CLAB).Furthermore, it is important to note that the measured values of acceleration can be influenced by various factors, such as CLAB precision of 3%, the complexity of aircraft manoeuvers, aircraft vibrations, inclination of the CLAB and the wall where the experiment is conducted.
For the other sequences the results are similar and, therefore, are not presented here.Figure 7 compares the acceleration obtained by the automatic system of the aircraft with the acceleration obtained by the CLAB accelerometer data for approximately 1000 s of the  The first three clusters of points correspond to the conditions of microgravity, lunar gravity and Martian gravity, respectively.The main cluster is all about the gravitational forces that occur during horizontal flight.It focuses on the vertical acceleration conditions between parabolic manoeuvres, with acceleration ranging from 0.9g to 1.3g.The cluster with the highest acceleration values corresponds to the hypergravity moments.
Figure 8 displays the aircraft altitude (in red) and the corresponding gravity acceleration (in black) for the first sequence of parabolas.The data was obtained through the aircraft's automatic data collection system during simulations of Martian and lunar gravities.
Considering g = 9.80 m s −2 , from the acceleration data in figure 8, the mean value of the Martian gravity is 3.72 ± 0.24 m s −2 , and for lunar gravity is 1.59 ± 0.08 m s −2 and 1.62 ± 0.06 m s −2 .The simulation times for Martian and lunar gravity were 30 s and 23 s,  Figure 9 displays the aircraft altitude (in red) and corresponding acceleration (in black) during microgravity conditions for the second sequence of parabolas.The data analysed for each parabola varies in time between 18.7 and 20.0 s.The mean values of acceleration, for the four parabolas, range between −0.02 ± 0.12 and 0.01 ± 0.12 m s −2 .
Figure 10 shows the aircraft during a parabolic manoeuvre, immediately after pull-up.The gravity acceleration felt inside the aircraft is the acceleration of gravity in the aircraft's frame of reference, ¢  g , which is given by AE and all objects in the aircraft's frame of reference are subject to the weight where  g is the acceleration of gravity in the Earth's reference frame at flight altitude, and  a AE is the acceleration of aircraft relative to the Earth.Note that - a AE is the inertial acceleration (see right picture of figure 10) [19].Based on the altitude of the aircraft over time, as shown in figure 4, the equation of motion y(t) during the parabola phase is a quadratic function.To achieve a true state of free fall, the pilots must neutralise all the forces acting on the aircraft (aerodynamic lift, aerodynamic drag and engine thrust) in addition to the force of gravity [1].Under the aforementioned conditions and considering that the aircraft's trajectory is flat, in a first approximation, the law of motion for the vertical position of the aircraft can be written like where y 0 and v y 0 are, respectively, the initial altitude and velocity at instant t .0 From the equations of the fit curves to the altitude data, in figure 8, as well as from the equations (3)-( 4), we can obtain the values of the acceleration of aircraft relative to the Earth, a .Taking into account that the Earth's acceleration of gravity at altitude of 8 km is = g −9.78 m s −2 [20], the acceleration of gravity felt inside the aircraft, ¢  g , can be determined by equation (1).Therefore, for the Martian parabola, ¢ g is −3.96 m s −2 and for the lunar parabolas ¢ g is −1.87 m s −2 and −1.85 m s −2 .These values, when compared with the acceleration calculated from the data collected by the aircraft's automatic system, differs 0.24 m s −2 for the Martian gravity and 0.28 m s −2 and 0.23 m s −2 for the lunar gravity (figure 8).
The difference between the average values of the accelerations obtained directly from the data collected by the accelerometer, and the accelerations determined from the law of motion, can be explained by the approximations inherent to the law of motion, equation (3), such as neglecting air resistance, the aircraft's automatic data acquisition system not being located at the aircraft's center of mass, the complexity of the aircraft manoeuvers required to simulate microgravity, Martian gravity and lunar gravity, etc.

Conclusions
This paper analyses the acceleration data recorded during a parabolic flight that took place in Beja, Portugal in September 2022.Acceleration data within the aircraft were collected by both the aircraft's automatic system and the CLAB mounted on the aircraft wall that separates the cockpit from the area designated for passengers.
Different levels of gravity were measured, from microgravity to hypergravity, including the g-levels of the Moon and Mars, based on direct readings of acceleration data.
From the adjustment of quadratic curves to altitude data as a function of time, the laws of the aircraft's vertical movement were extracted.
The accelerations obtained by the aircraft's automatic data acquisition system and by the experiment set up in the aircraft and carried out during the flight are in good agreement.However, the Martian gravity, lunar gravity and microgravity determined from the laws of vertical movement of the aircraft differ from the gravities calculated from the data collected by the aircraft's automatic system within the range of 0.23-0.38m s −2 .
This paper has implications on at least three levels: firstly, the authors believe students would benefit if they were to plan activities to be performed during parabolic flights, since such experiences can help promote students' interest in STEM (science, technology, engineering, and mathematics), namely in areas linked to space; secondly, carrying out activities in microgravity environment, analysing and processing data collected from research of this kind, can serve to bring science learned at school closer to the scientific community; finally, this work can serve to introduce the study of inertial forces in secondary and higher education.
As future work, the authors believe it would be beneficial that the data collected by the force sensor could be used to study force as a function of acceleration, as well as to explore Newton's second law in parabolic flight conditions and, consequently, improve the students' understanding of the equivalence principle, which states that 'inertial force and gravitational force are the same thing' [19,21].

Figure 1 .
Figure 1.Participants inside of Airbus A310 after a parabolic flight in Beja in 2022.Photo provided by Novespace [11].

Figure 4 (
Figure 4(a) displays four sequences of g-parabolas (1, 2, 3, and 4) obtained from the aircraft's automatic data.The black lines represent the acceleration normalized (a/g) to the Earth's gravity (g = 9.80 m s −2 ), and the red lines represent the altitude profiles recorded during the parabolic flight.The data acquisition was performed at 0.1 s intervals.During the first sequence (figure 4(b)), the aircraft executed one parabola with Mars and two with Moon acceleration.In the second and third sequences (figures 4(c) and (d), respectively), the aircraft performed four 0g-parabolas.The fourth sequence (figure 4(e)) consisted of five 0g parabolas.The typical acceleration provided by the aircraft's recording system during the parabolas was measured using microaccelerometer attached to the cabin floor structure.Figures 4(a) and (b) show that the parabolas in sequence 1 had a duration of reduced gravity of approximately 30 s for Martian and 20 s for lunar acceleration, with accelerations of around 0.38g and 0.16g, respectively (refer to figure 4(b)).Meanwhile, the parabolas in sequences 2, 3, and 4 had a period of microgravity between 18 and 20 s, as demonstrated in figures 4(c)-(e).The altitude range during the first sequence, (Mars and Moon parabolas) was, approximately, between 8.1 and 8.3 km (see figure 4(b)).For the second, third, and fourth sequences (the microgravity parabolas), the altitude ranged approximately between 8.5 and 8.6 km (see figures 4(c)-(e)).Data acquired by CLAB accelerometerThe acceleration components (ax, ay, az) measured using CLAB are displayed as a function of time in figure5.It is possible to observe the following: (a) ax (the acceleration in the lateral direction of the aircraft axis) remains almost constant throughout the aircraft's manoeuvres, with an average value and standard deviation of ax = −0.02± 0.06 m s −2 (mean ± SD); (b) ay (the acceleration in the vertical direction of the aircraft axis) is the main acceleration felt by passengers during airplane manoeuvres, and it varies between microgravity and hypergravity (∼1.8g);(c) az (the acceleration in the direction of the aircraft's longitudinal axis) presents variations between −0.40 and 2.35 m s −2 and the highest values are observed in hypergravity phases.

Figure 4 .
Figure 4. Acceleration normalized to Earth's gravity (black line) and altitude (red line) profiles recorded during the parabolic flight.Profiles for: (a) complete parabolic flight between 1500 and 6000 s; (b) first sequence of parabolas; (b) second sequence of parabolas; (c) third sequence of parabolas; (d) fourth sequence of parabolas.The Earth's gravity is g = 9.80 m s −2 .

Figure 5 .
Figure 5. Acceleration components obtained from CLAB accelerometer data for approximately 1000 s. ax-green line, ay-black line, az-red line.

Figure 6 .
Figure 6.Comparison of aircraft's automatic acceleration normalized to Earth's gravity (black line) and acceleration obtained from CLAB accelerometer data (green line) for approximately 1000 s of the parabolic flight.

Figure 7 .
Figure 7. Aircraft's automatic acceleration normalized to Earth's gravity versus acceleration obtain from CLAB accelerometer data for approximately 1000 s of the parabolic flight.Data was obtained at intervals of one second.

Figure 8 .
Figure 8. Acceleration normalized to Earth's gravity (black line) and altitude (red line) profiles recorded during the parabolic flight for the first sequence of parabolas.The fit curves to the altitude data are superimposed with the points.

Figure 9 .
Figure 9. Acceleration normalized to Earth's gravity (black line) and altitude (red line) profiles recorded during the parabolic flight for the second sequence of parabolics.The fit curves to the altitude data are superimposed with the points.

Figure 10 .
Figure 10.Acceleration of aircraft during parabolic flight after pull-up (left picture) and acceleration of gravity felt inside the aircraft (right picture). v AE is the velocity of the aircraft relative to Earth.Photos provided by Novespace and modified by the authors [11].

AE
Thus, for the Martian parabola, a AE is −5.818 m s −2 and for the lunar parabola, a AE is −7.909 m s −2 and −7.932 m s −2 , (figures 4(b) and 8).During the microgravity parabolas the values of a AE are −9.524m s −2 , −9.425 m s −2 , −9.423 m s −2 and −9.404 m s −2 , respectively, for each one of the parabolas (figures 4(c) and 9).