Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails

Sérgio Sacani
Sérgio SacaniSupervisor de Geologia na Halliburton em Halliburton

With interstellar mission concepts now being under study by various space agencies and institutions, a feasible and worthy interstellar precursor mission concept will be key to the success of the long shot. Here we investigate interstellar-bound trajectories of solar sails made of the ultra lightweight material aerographite. Due to its extremely low density (0.18 kgm−3) and high absorptivity (∼1), a thin shell can pick up an enormous acceleration from the solar irradiation. Payloads of up to 1 kg can be transported rapidly throughout the solar system, e.g. to Mars and beyond. Our simulations consider various launch scenarios from a polar orbit around Earth including directly outbound launches as well as Sun diver launches towards the Sun with subsequent outward acceleration. We use the poliastro Python library for astrodynamic calculations. For a spacecraft with a total mass of 1 kg (including 720 g aerographite) and a cross-sectional area of 104 m2, corresponding to a shell with a radius of 56m, we calculate the positions, velocities, and accelerations based on the combination of gravitational and radiation forces on the sail. We find that the direct outward transfer to Mars near opposition to Earth results in a relative velocity of 65 kms−1 with a minimum required transfer time of 26 d. Using an inward transfer with solar sail deployment at 0.6AU from the Sun, the sail’s velocity relative to Mars is 118 kms−1 with a transfer time of 126 d, whereMars is required to be in one of the nodes of the two orbital planes upon sail arrival. Transfer times and relative velocities can vary substantially depending on the constellation between Earth andMars and the requirements on the injection trajectory to the Sun diving orbit. The direct interstellar trajectory has a final velocity of 109 kms−1. Assuming a distance to the heliopause of 120AU, the spacecraft reaches interstellar space after 5.3 yr. When using an initial Sun dive to 0.6AU instead, the solar sail obtains an escape velocity of 148 kms−1 from the solar system with a transfer time of 4.2 yr to the heliopause. Values may differ depending on the rapidity of the Sun dive and the minimum distance to the Sun. The mission concepts presented in this paper are extensions of the 0.5 kg tip mass and 196m2 design of the successful IKAROS mission to Venus towards an interstellar solar sail mission. They allow fast flybys atMars and into the deep solar system. For delivery (rather than fly-by) missions of a sub-kg payload the biggest obstacle remains in the deceleration upon arrival.

arXiv:2308.16698v1
[astro-ph.IM]
31
Aug
2023
Ultrafast transfer of low-mass payloads to Mars and beyond using
aerographite solar sails
Julius Karlapp a,∗
, René Heller b,1
and Martin Tajmar a
aInstitute of Aerospace Engineering, Technische Universität Dresden, Marschnerstr. 32, 01307 Dresden, Germany
bMax Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany
A R T I C L E I N F O
Keywords:
aerographite
solar sails
Mars transfer
interstellar precursor
A B S T R A C T
With interstellar mission concepts now being under study by various space agencies and institutions,
a feasible and worthy interstellar precursor mission concept will be key to the success of the long
shot. Here we investigate interstellar-bound trajectories of solar sails made of the ultra lightweight
material aerographite. Due to its extremely low density (0.18 kg m−3
) and high absorptivity (∼1), a
thin shell can pick up an enormous acceleration from the solar irradiation. Payloads of up to 1 kg can
be transported rapidly throughout the solar system, e.g. to Mars and beyond. Our simulations consider
various launch scenarios from a polar orbit around Earth including directly outbound launches as well
as Sun diver launches towards the Sun with subsequent outward acceleration. We use the poliastro
Python library for astrodynamic calculations. For a spacecraft with a total mass of 1 kg (including
720 g aerographite) and a cross-sectional area of 104 m2, corresponding to a shell with a radius of 56 m,
we calculate the positions, velocities, and accelerations based on the combination of gravitational and
radiation forces on the sail. We find that the direct outward transfer to Mars near opposition to Earth
results in a relative velocity of 65 km s−1
with a minimum required transfer time of 26 d. Using an
inward transfer with solar sail deployment at 0.6 AU from the Sun, the sail’s velocity relative to Mars
is 118 km s−1
with a transfer time of 126 d, where Mars is required to be in one of the nodes of the two
orbital planes upon sail arrival. Transfer times and relative velocities can vary substantially depending
on the constellation between Earth and Mars and the requirements on the injection trajectory to the Sun
diving orbit. The direct interstellar trajectory has a final velocity of 109 km s−1
. Assuming a distance
to the heliopause of 120 AU, the spacecraft reaches interstellar space after 5.3 yr. When using an
initial Sun dive to 0.6 AU instead, the solar sail obtains an escape velocity of 148 km s−1
from the
solar system with a transfer time of 4.2 yr to the heliopause. Values may differ depending on the
rapidity of the Sun dive and the minimum distance to the Sun. The mission concepts presented in this
paper are extensions of the 0.5 kg tip mass and 196 m2 design of the successful IKAROS mission to
Venus towards an interstellar solar sail mission. They allow fast flybys at Mars and into the deep solar
system. For delivery (rather than fly-by) missions of a sub-kg payload the biggest obstacle remains in
the deceleration upon arrival.
1. Introduction
From the first medium distance ride of a steam locomo-
tive in 1804, the ‘Penydarren’ of British mining engineer
Richard Trevithick, at about 4 km hr−1
to the current helio-
centric escape velocity of the Voyager 1 probe at 17 km s−1
,
humanity has managed a velocity gain of their mechanical
and automated vehicles of over four orders of magnitude
within about 200 yr [1]. If technological progress continues
to increase humanity’s top velocitys at this rate, that is,
a doubling every 15 yr for about another seven doublings,
we would reach 1 % of the velocity of light (𝑐) within
little more than 100 yr from now. At this point, our travel
to the stars might become reality. With our closest stellar
neighbor Proxima Centauri located at a distance of about
1.3 pc (4.2 ly) [2] a velocity of 0.1 𝑐 implies a travel time
of about 42 yr, well within a human lifetime and within the
46 yr of operation time of the Voyager1 mission.
Recent studies suggest that the technology and knowl-
edge components for interstellar missions are in reach, such
as miniaturized electronic devices [3], advanced information
processing [4, 5], ultra-light materials such as graphene
∗ Corresponding author (E-Mail: julius.karlapp@gmail.com).
1 Presenting author, 8th Interstellar Symposium 2023.
[6] and aerographite [7] that could be used for light sails,
and predictions of the light sail dynamics [8, 9, 10]. Solar
sail technology has been demonstrated to work during var-
ious missions in the solar system, such as IKAROS [11],
NanoSail-D1, and LightSail-2 [12] (see [10, 13] for further
references).
An intermediate step toward interstellar travel are inter-
stellar precursor missions [14], some of which have been
proposed for a comet rendevous [15] and deep solar system
exploration, e.g. the search for the suspected Planet Nine
[10, 16].
In this paper, we explore an interstellar precursor sce-
nario that we expect to be more relevant in the near future.
Driven by NASA’s goal to put humans on Mars by the end
of the 2030s [17] and related plans by the SpaceX company,
we explore the potential of solar lightsails to carry small
payloads to Mars in short times, that is, weeks to months.
The energy and thus monetary demands of conventional
chemical rocketry depend strongly on the Earth-Mars orbital
configuration. As a result, launches are usually chosen to
follow a low-energy Hohmann transfer orbit, the launch
window of which opens up every 780 d or 2.1 yr.
1 https://ntrs.nasa.gov/citations/20110015650
Karlapp et al.: Paper submitted to Acta Astronautica for peer review Page 1 of 8
Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails
Nomenclature
Δ𝑡 Duration
Δ𝑣 Change in velocity
𝜅rad Radiation coupling constant
 Absorptivity
𝜌 Aerographite density
𝜀 Shell thickness
𝑎 Acceleration
𝑎 Semi-major axis
𝑎max Maximum acceleration
𝑐 Speed of light
𝐹grav Gravitational force
𝐹rad Force of solar radiation
𝐹tot Force applied to the sail
𝑔 Earth surface gravity
ℎminimum Minimum orbit height
𝑖 Inclination
𝑙 Sail radius
𝐿⊙ Solar luminosity
𝑚 Spacecraft mass
𝑀⊙ Solar mass
𝑚aero Aerographite mass
𝑀p Planets mass
𝑟 Solar distance
𝑟Hill,Earth Hill radius of Earth
𝑟Hill,Mars Hill Radius of Mars
𝑟Hill Hill radius
𝑅⊕ Radius reference body
𝑆 Sail cross sectional area
𝑡 time
𝑡deploy Time sail deployment
𝑡end Final transfer time
𝑡entry Entry time
𝑡exit,Earth Exit time from Earth’s sphere of influence
𝑡Mars Duration inside martian sphere of influence
𝑣 Velocity
𝑣Earth Orbital velocity of Earth around the Sun
𝑣end Final transfer velocity
𝑣entry Entry velocity
𝑣exit,Earth Exit velocity from Earth’s sphere of influ-
ence
These time scales have been proven manageable for fully
robotic missions and even for more longterm missions to
Mars like the Mars sample return, which are effectively a
sequence of Mars missions that build on each other. Once the
human factor comes into play, however, risk and emergency
assessments change substantially and a wait time of months
to years, e.g. for medical supply or replenishmentof essential
materials or devices may be too long.
In this study we feed, for the first time, results from
laboratory studies on the material properties of aerographite
[7] into solar sail trajectory simulations that aim at Mars
and beyond. Aerographite, with its extremely low density
of 0.18 kg m−3
(compared to 2700 kg m3
for aluminum foil,
for example) and a radiation coupling constant 𝜅rad ∼ 1
that permits very efficient transformation of solar irradiation
to acceleration, stands out as a material for solar sail ap-
plications [10]. Being a carbon-based aerogel, aerographite
can be described as a foam-like material with a microscopic
structure composed of 𝜇m-sized carbon microtubes [18, 19,
20]. Importantfor the structural integrity of any macroscopic
application, aerographite is characterized by a high elas-
ticity. Elastic deformation occurs up to a compression rate
of 95%. The combination of low density, efficient radiative
coupling,and high elasticity makes aerographitean excellent
candidate material for solar sails.
Beyond solar sail concepts to Mars, we study the dynam-
ical properties of aerographite sails that reach escape veloci-
ties from the solar system. Hence, solar sail missions to Mars
can serve as natural precursors for interstellar missions to
establish the technologies for long-distance communication,
deep space navigation etc.
2. Methods - Simulation Script
To investigate interplanetary and interstellar transfers and to
simulate possible trajectory progressions we use the Python
programminglanguage in combination with the poliastro li-
brary for astrodynamic calculations and visualization 2. Our
objective is to use numerical calculations of the positions,
velocities, and accelerations for various sail trajectories us-
ing adaptive time steps.
The force on our hypothetical sail is parameterized as
𝐹tot = 𝐹rad + 𝐹grav =
1
𝑟2
( 𝐿⊙
4𝜋𝑐
𝑆𝜅rad − 𝐺𝑀𝑚
)
, (1)
where 𝑟 is the solar distance, 𝐿⊙ the solar luminosity, 𝑆
the sail cross sectional area, 𝑀 is the mass of the primary
source of gravity for the sail that defines the current sphere
of influence, and 𝑚 the spacecraft mass [10].
We consider an aerographite sail with 𝜅rad = 1 that is
shaped as a hollow sphere with a cross-sectional area of
𝑆 = (100 m)2 = 104 m2. This corresponds to a sail radius
of 𝑙 =
√
𝑆∕𝜋 = 56.4 m. We assume spherically symmetric
distribution of the sail mass with a total mass of 𝑚 = 1 kg.
Although our equations are ignorant as to the contribution
2 https://docs.poliastro.space/en/stable/
Karlapp et al.: Paper submitted to Acta Astronautica for peer review Page 2 of 8
Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails
of the aerographite and the payload to the total mass of the
spacecraft, we can use Eq. (11) from Heller et al. [10] to
estimate the aerographite mass 𝑚aero = 𝜋𝑙24𝜌𝜀, where 𝜀 is
the shell thickness. For a nominal 0.1 mm thin hollow sphere
with a radius of 56.4 m this gives 𝑚aero = 720 g, leaving
280 g for payload. Using a hollow aerographite hemisphere
would halve the aerographite mass and permit 640 g of
payload.
During interplanetary transfer we assume a two-body
problem of the sail and its dominant gravitational counter-
part, the latter of which can be the Sun, the Earth or Mars in
our case. As long as the sail is beyond the sphere of influence
(SOI) of a planet, we apply Eq. (1) with 𝑀 = 𝑀⊙, or else
we substitute the solar mass with the mass of the respective
planet. We estimate the planetary SOI via the Hill radius
𝑟Hill = 𝑎 ⋅
3
√
𝑀p
3𝑀⊙
, (2)
inside of which the gravitational attraction of the planet on
a point mass (i.e. the sail) dominates over the gravitational
force from the Sun. The Hill radius is defined by the solar
mass, the mass of the planet 𝑀p, and the semi-major axis (𝑎)
of the Sun-planet orbit, which is approximated to be circular.
The Earth’s Hill radius is 𝑟Hill,Earth ≈ 1.5 × 106 km, while
the martian Hill radius is 𝑟Hill,Mars ≈ 1.1 ×106 km [21, 22].
3. Results
In Table 1 we summarize our setup of the possible launch
scenarios. We consider both direct outward transfers and
inward transfers. The outward injection calculates for a
launch from a parking orbit around Earth radially away
from the Sun (initially defined in the gravitational sphere
of influence of Earth), while the inward injection calculates
for an initial conventional launch towards the Sun with a
delayed sail deployment in a lower solar orbit (Sun diver
approach, initially defined in the solar gravitational sphere
of influence).
3.1. Outward Trajectories
We now investigate the ability of our hypothetical space-
craft to reach Mars or interstellar space by calculating the
spacecraft position, velocity, and acceleration as a function
of time.
3.1.1. Outward Mars Transfer
For a direct outwards transfer from Earth to Mars, Mars
needs to be near opposition since the vector of the solar
radiation pressure will push the spacecraft radially away
from the Sun. Thereforetwo possible starting orbits (parking
orbits) around Earth are conceivable, a High Elliptical Orbit
(HEO) or a Polar Orbit (PO) [23]. Previous investigations
showed that other standard orbits tend to produce collisions
with Earth or erratic behavior of the spacecraft [10, Sect. 3
therein].
Table 1
General orbit configurations for heliocentric injections of a solar
sail spacecraft (𝑚 = 1 kg, 𝐴 = 104
m2
). All outward injections are
referring to the geocentric reference system (GCRS) because the
parking orbit is defined around Earth, whilst the inward injections
are defined in the heliocentric reference system (HCRS) because
the spacecraft is simulated directly in the solar gravitational sphere.
Heliocentric Injection Outward (GCRS) Inward (HCRS)
Destination Mars Interstellar Mars Interstellar
Orbit Polar Polar Solar Solar
semi-major axis [𝑘𝑚] 42000 42000 149.6 ⋅ 106
149.6 ⋅ 106
Inclination [◦
] 113.4 113.4 0 0
Eccentricity [−] 0.24 0.24 0.0167 0.0167
For an interplanetary transfer mission the use of a polar
orbit is preferable (Table 1). Due to the orientation of the
polar orbit the orbital plane of the spacecraft is oriented
perpendicular to the solar radiation vector as shown in
Fig. 1. Therefore every point along the orbit experiences
a continuous irradiation from the sun (the Earth’s shadow
does not matter). A time independent deployment of the
sail is possible. Referring to the geocentric reference system
the inclination angle of the orbit plane is 113.4◦ due to
the tilt of the Earth’s spin axis, whereas in the heliocentric
reference system the inclination is 90◦. The semi-major axis
of the PO is chosen to be 42000 km. For smaller values the
produced trajectories wont overcome the gravitational pull
of the Earth. For larger values the gain in net acceleration
is marginal. Since the altitude of the PO around the Earth
is negligible compared to the scale of an astronomical unit
(PO altitude is 2.4 × 10−4 AU) the magnitude of the solar
radiation pressure can be assumed to be constant.
Figure 1 shows the PO (black) around Earth (blue),
chosen to calculate the outward injections following Table 1.
The escaping trajectory (light blue) can be seen to align with
the vector of the solar radiation pressure (red arrow) over
long distances.
Upon deployment of the sail (e.g. unfolding in space) in
the PO, the trajectories initially spiral away from Earth due
to the two acting forces of the solar radiation pressure and
gravitational force of Earth. As the sail picks up speed, its
trajectory resembles more and more a straight line. Any live
correction of the trajectory would need additional attitude
control mechanism, which we neglect in our simulations.
Instead, we consider the sail a projectile under the combined
photogravitational effects.
The sail initially follows a curly trajectory as shown in
Fig. 1. The Earth’s gravitational pull reduces the departing
velocity by about 1.5 km s−1
over the course of about 1 d
(Fig. 2). Thereafter, the force from the solar irradiation takes
over and induces a net acceleration of the sail. At the time
the sail enters interplanetary space after 3.54 d the spacecraft
has an exit velocity from the Earth’s SOI of 11.2 km s−1
. 3
After the transition from the SOI of Earth to interplan-
etary space, that is, into the Sun’s SOI the spacecraft enters
3The identity with the escape velocity from Earth is pure coincidence.
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Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails
Earth SOI Outward Trajectory
Earth
Vector of Solar Radiation Pressure
Parking Orbit
Trajectory
−8 0 8 16 24 32
Distance x
[
R⊕
]
−8
0
8
16
24
32
Distance y [
R
⊕
]
−8
0
8
16
24
32
40
Distance
z
[
R
⊕
]
Figure 1: The polar orbit (PO, black) with the escaping
trajectory (light blue) for a direct outward ejection. The PO
is in the 𝑦-𝑧 plane. The solar illumination (red arrow) is along
the 𝑥 axis.
0
2
4
6
8
10
12
0 0.5 1 1.5 2 2.5 3 3.5 4
transition
to
solar
sphere
of
inf
luence
Velocity
[
km
s
−1
]
Time [d]
Outward Transfer
Figure 2: Diagram of the spacecraft velocity 𝑣 as a function of
time 𝑡 inside Earth’s SOI for an outward transfer.
the martian SOI. Inside the martian SOI several trajectories
are presented (Fig. 3). Those trajectories only differ slightly
in terms of their starting point from the initial parking orbit
around Earth. For the purpose of a better visual inspection
these trajectories are shown in the vicinity of Mars. Any
trajectory that reaches an altitude of 200 km or less above the
martian surface is considered a collision (light blue) while
the other trajectories (grey) escape the martian SOI.
The spacecraft enters the martian SOI 613.8hr or 25.6 d
after launch from the Earth’s PO (Fig. 4). After just another
4.5 hr inside the martian SOI, the spacecraft reaches Mars.
During the flight inside the martian SOI the spacecraft
−10
−5
0
5
10
Distance x
[
R⊕
]
−10
−5
0
5
10
Distance y [
R
⊕
]
−8
−6
−4
−2
0
2
4
6
8
Distance
z
[
R
⊕
]
Mars SOI Trajectory Scatter
Mars
FlyBy Trajectory Scatter
Collision Trajectory Scatter
Figure 3: Three-dimensional view of several trajectories inside
the martian SOI. Trajectories in light blue color collide with
Mars, trajectories in grey color denote fly-bys.
65.3
65.4
65.5
65.6
65.7
65.8
65.9
613 614 615 616 617 618 619
transition
to
martian
sphere
of
inf
luence
reaching
martian
orbit
Velocity
[
km
s
−1
]
Time [hr]
Outward Transfer
Figure 4: Diagram of the spacecraft velocity 𝑣 as a function of
time 𝑡 inside the martian SOI for an outward transfer.
is subject to an acceleration by the martian gravity and
therefore gains 0.51 km s−1
which gives it a final velocity
of 65.9 km s−1
.
3.1.2. Outward Interstellar Transfer
Regarding the outward interstellar transfer we used the same
polar parking orbit around Earth to deploy the spacecraft
(Table 1). Instead of targeting Mars, however, all trajectories
have been extended until they reached the boundary of
the solar system, which we define at the distance of the
heliopause at 𝑟 = 120 AU.
The initial acceleration in the Earth’s orbit corresponds
to the maximum acceleration of 0.33 m s−2 (dashed line in
Fig. 5), corresponding to about 3 %𝑔. The acceleration then
decreases following the inverse square law as a function of
distance from the Sun. The factor applies also for gravita-
tional forces and was added to the general thrust equation
(Eq. 1).
Karlapp et al.: Paper submitted to Acta Astronautica for peer review Page 4 of 8
Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.01 0.1 1 10 100 1000
sail deployment
sail deployment
Earth
′
s
sphere
of
inf
luence
solar
sphere
of
inf
luence
Acceleration
[
m
s
−2
]
Time [d]
Outward Transfer
Inward Transfer
Figure 5: Diagram of the spacecraft acceleration 𝑎 as a function
of time 𝑡 for both outward and inward transfers.
0
20
40
60
80
100
120
140
160
0.01 0.1 1
outward
transfer
time
to
interstellar
inward
transfer
time
to
interstellar
Velocity
[
km
s
−1
]
Time
[
yr
]
Outward Transfer
Inward Transfer
Figure 6: Diagram of the spacecraft’s velocity 𝑣 as a function
of time 𝑡 in interplanetary space.
As a result, the velocity of the spacecraft increases
continuously, but ever less with increasing distance to the
Sun. The spacecraft ultimately reaches a terminal velocity
of 𝑣end = 109 km s−1
after a duration of about 2 yr (Fig. 6).
At this maximum speed, the spacecraft reaches the boundary
of the solar system after 𝑡end = 5.3 yr.
We executed a limited series of trajectory simulations
with variations of the launch configurations from the initial
parking orbit around the Earth and found none that yielded
any significant increase of the terminal velocity. The ter-
minal velocity of the spacecraft can only be increased by
reducing its overall mass or by increasing its cross sectional
surface.
3.2. Inward Trajectories
An alternative approach to increase the terminal velocity of
the spacecraft is to reduce the distance to the Sun for the
initial sail deployment. The increased photon pressure in
−2
−1.5
−1
−0.5
0
0.5
1
Distance x [AU]
−2
−1.5
−1
−0.5
0
0.5
1
D
istance y
[AU
]
−1.5
−1
−0.5
0
0.5
1
1.5
Distance
z
[AU]
Possible Directions of Interstellar Trajectories
Sun
Earth
Δvbreaking = 4.5 km s−1
6.0
7.6
9.1
10.6
12.1
13.6
15.1
16.6
Trajectory Scatter
Earth Orbit
Minimum Distance
Figure 7: The Sun diver approach allows the spacecraft to
achieve several radial trajectory directions inside Earth’s orbital
plane around the Sun. The required breaking maneuver in
the parking orbit around Earth is labeled Δ𝑣breaking for each
direction. Steeper maneuvers require a higher amount of Δ𝑣.
the vicinity to the Sun results in a higher radiation force
component and therefore a shorter transfer time than the
direct outward approach from Earth.
To prove this thesis our script was modified to simulate
a two phase flight. For the first phase a conventional impulse
maneuver using chemical engines in an orbit around Earth
towards the Sun is executed, lowering the spacecraft’s orbit
around the Sun (Table 1). Once it reaches a predefined solar
distance of 0.6 AU (black circle in Fig. 7), we assume that
the sail is deployed, which initiates the second flight phase.
From that point on, the spacecraft accelerates radially out-
ward and therefore is pushed away from the Sun. We tested
this method both for a transfer to Mars and to interstellar
trajectories.
In addition to an increase of the terminal velocity this
Sun diver approach offers another advantage. By adjusting
the breaking velocity from the parking orbit around Earth
the rapidity of the fall towards the Sun can be controlled.
In Fig. 7 we present the trajectory scatter for various val-
ues of Δ𝑣breaking. The Δ𝑣 was calculated considering the
spacecrafts standard configuration with respect to the initial
velocity of Earth’s orbit (𝑣Earth = 29.78 km s−1
). Steeper
dives require stronger breaking. By applying the correct
amount of Δ𝑣breaking or deploying the sail at the second
intersection between the initial trajectory and the minimum
distance every direction is possible inside Earth’s orbital
plane.
To implementthe Sun divermaneuversome assumptions
had to be made. To minimize the required Δ𝑣 for the break-
ing maneuverat Earth,the minimumdistance aroundthe Sun
(the deploymentaltitude) must not be too low. Yet, in orderto
reduce the transfer time and to exploit the advantages of the
first-in-then-out approach the minimum distance needs to be
sufficiently low. Furthermore, for a single impulse maneuver
in Earth’s SOI, Mars needs to be located in the leading
Karlapp et al.: Paper submitted to Acta Astronautica for peer review Page 5 of 8
Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails
−2 −1.5 −1
−0.5
0 0.5 1 1.5 2
Distance x [AU]
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
D
istance y
[AU
]
−1
−0.5
0
0.5
1
1.5
Distance
z
[AU]
Inward Mars Transfer
Sun
Earth
Mars
Position Earth on Arrival
Position Mars at Launch
Trajectory
Earth Orbit
Mars Orbit
Minimum Distance
Figure 8: Solar system showing Earth’s and Mars’ orbit around
the Sun. The calculated trajectory for the inward Mars transfer
reaches Mars in the leading node of both orbital planes.
node of the two orbital planes upon arrival of the space-
craft. Multi-impulse maneuvers are also possible but have
not been tested for this publication. With a multi-impulse
maneuver the inclination change can be compensated and
furthermorethe range of possible trajectories around the Sun
be increased.
3.2.1. Inward Mars Transfer
For the Mars transfer a random constellation between Earth
and Mars was generated. As a boundary condition, however,
we searched for trajectories from Earth that lead to encoun-
ters with Mars when Mars is in the ascending node of the two
orbital planes. In these cases the spacecraft does not need
to leave the ecliptic and we can consider the problem in a
plane. To inject the spacecraft towards the Sun, a negative
impulse of Δ𝑣 ≈ 4.5 km s−1
needs to be applied in the
parking orbit around Earth. The minimum solar distance
for sail deployment was again set to 0.6 AU. Our numerical
script contains two loops referring to the solar SOI as well as
the martian SOI. In Fig. 8 the solar system is shown with the
additional orbit of Mars (orange) next to the orbit of Earth
(blue). Both the position of Earth on arrival and the position
of Mars at launch are shown faded. The required trajectory
to reach Mars is presented in green.
Using the standard spacecraft configuration we calcu-
late a total transfer time of 125.6 d or 4.2 months (Fig. 9).
When reaching the martian orbit the spacecraft has a ve-
locity of 118.02 km s−1
. Considering the entry velocity of
≈ 117.85 km s−1
, we find that it gained Δ𝑣 = 0.17 km s−1
inside the martian SOI. Due to the high velocity the time
spent in the martian SOI is just Δ𝑡Mars ≈ 2.5 hr.
In summary, the sail was deployed after it reached the
minimum solar distance of 0.6 AU about 103 d after launch.
The maximum acceleration was determined as roughly
0.11 m s−2.
117.8
117.85
117.9
117.95
118
118.05
3012.5 3013 3013.5 3014 3014.5 3015 3015.5
Velocity
[
km
s
−1
]
Time [hr]
Inward Transfer
transistion
to
martian
sphere
of
inf
luence
reaching
martian
orbit
Figure 9: Diagram of the spacecraft’s velocity 𝑣 as a function
of time 𝑡 inside the martian SOI for an inward transfer.
−1
−0.5
0
0.5
1
Distance x [AU]
−1
−0.5
0
0.5
1
D
istance y
[AU
]
−1
−0.5
0
0.5
1
Distance
z
[AU]
Inward Interstellar Trajectory
Sun
Earth
Trajectory
Earth Orbit
Minimum Distance
Figure 10: Solar system showing Earth’s orbit around the Sun.
The calculated trajectory for the inward interstellar transfer
escapes the solar system in a straight line.
3.2.2. Inward Interstellar Transfer
For the inward interstellar transfer the same approach was
used as in subsection 3.2.1 for the Sun diver maneuver to
Mars (Table 1). Starting in a solar orbit equally to the Earth’s
orbit the orbit of the spacecraft is lowered by initiating a neg-
ative impulse maneuver. By reaching the minimum_distance
the sail is deployed and the solar radiation pressure pushes
the spacecraft away from the Sun. Yet, instead of stopping
the calculating function when reaching the Mars SOI the tra-
jectories are continued. As in subsection 3.1.2 the function
then is stopped when the spacecraft reaches a distance of
𝑟 = 120 AU. Again the trajectories showed a linear progres-
sion after performing the Sun diver maneuver in the close
vicinity of the Sun (Fig. 10).
As shown in Fig. 5 a spacecraft with standard con-
figuration on an inward interstellar transfer experiences a
maximum acceleration of 0.11 m s−2 occurring after 103 d.
Karlapp et al.: Paper submitted to Acta Astronautica for peer review Page 6 of 8
Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails
After the sail deployment the velocity increases again to its
maximum of 148 km s−1
(Fig. 6). This maximum velocity is
reached after about 300 d. Flying with its maximum velocity
the spacecraft reaches the boundary of the solar system after
roughly 1520 d or 𝑡end = 4.16 yr.
Even closer solar approaches than those at 0.6 AU that
we tested are certainly possible. For context, the still active
Parker Solar Probe has previously survived several hours
interior to 0.1 AU and as close as 0.086 AU from the Sun
[24].
4. Conclusions
We find that a 1 kg solar sail with a cross-sectional area
of 𝑆 = (100 m)2 = 104 m2 (radius of 56 m) can be
accelerated to a velocity of 𝑣entry = 65 km s−1
relative to
Mars in a direct outward transfer from Earth (when Mars
is near opposition). In this scenario, Mars is reached 26 d
after launch. An initial Sun diver maneuver down to 0.6 AU
from the Sun increases the travel time and flyby velocities to
Mars but allows launches from various Earth-Mars orbital
constellations.
Our simulations of direct launches to an interstellar tra-
jectory reveal terminalescape velocities of 𝑣end = 109 km s−1
and passage through the heliopause after 5.3 yr. If instead
the sail is first directed to the Sun, then a final velocity
of 𝑣end = 148 km s−1
is obtained with a transfer time of
𝑡end = 4.2 yr to the heliopause. Details depend on how
fast the sail is directed to the Sun (e.g. using conventional
chemical vehicles) and how close its perihelion distance.
The trajectories and solar sail properties presented in this
paper can be considered as a next step after the successful
IKAROS mission to Venus, which had a comparable mass
(0.5 kg) but smaller sail area 196 m2. The key difference is in
our assumption of aerographite as a sail material compared
to the polyimide sheet of IKAROS, which had a mass
of about 0.01 kg m−2
compared to 7.6 × 10−7 kg m−2
for
aerographite.
An as of yet unresolved question about our concept is the
decelerationof the payloadif this is supposed to be a delivery
and not a flyby mission. One possible scenario that we
suggest for future studies is the deceleration using friction
with the Martian atmosphere. Once dropped on the martian
surface, radioactive contamination of the payload can help to
recovereven gram-sizedpayloadon the Martian surface.The
key challenges would, of course, be the structural integrity of
the payload (not necessarily of the sail) during descent given
the expected high acceleration and temperature. But the low
gravity and the extended atmosphere of Mars actually yield
a relatively wide deceleration corridor and moderate heating
rates compared to Earth and Venus, making Mars a plausible
target for aerocapture [25].
Acknowledgements: R. H. acknowledges support from
the German Aerospace Agency (Deutsches Zentrum für
Luft- und Raumfahrt) under PLATO Data Center grant
50OO1501.
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Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails

  • 1. arXiv:2308.16698v1 [astro-ph.IM] 31 Aug 2023 Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails Julius Karlapp a,∗ , René Heller b,1 and Martin Tajmar a aInstitute of Aerospace Engineering, Technische Universität Dresden, Marschnerstr. 32, 01307 Dresden, Germany bMax Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany A R T I C L E I N F O Keywords: aerographite solar sails Mars transfer interstellar precursor A B S T R A C T With interstellar mission concepts now being under study by various space agencies and institutions, a feasible and worthy interstellar precursor mission concept will be key to the success of the long shot. Here we investigate interstellar-bound trajectories of solar sails made of the ultra lightweight material aerographite. Due to its extremely low density (0.18 kg m−3 ) and high absorptivity (∼1), a thin shell can pick up an enormous acceleration from the solar irradiation. Payloads of up to 1 kg can be transported rapidly throughout the solar system, e.g. to Mars and beyond. Our simulations consider various launch scenarios from a polar orbit around Earth including directly outbound launches as well as Sun diver launches towards the Sun with subsequent outward acceleration. We use the poliastro Python library for astrodynamic calculations. For a spacecraft with a total mass of 1 kg (including 720 g aerographite) and a cross-sectional area of 104 m2, corresponding to a shell with a radius of 56 m, we calculate the positions, velocities, and accelerations based on the combination of gravitational and radiation forces on the sail. We find that the direct outward transfer to Mars near opposition to Earth results in a relative velocity of 65 km s−1 with a minimum required transfer time of 26 d. Using an inward transfer with solar sail deployment at 0.6 AU from the Sun, the sail’s velocity relative to Mars is 118 km s−1 with a transfer time of 126 d, where Mars is required to be in one of the nodes of the two orbital planes upon sail arrival. Transfer times and relative velocities can vary substantially depending on the constellation between Earth and Mars and the requirements on the injection trajectory to the Sun diving orbit. The direct interstellar trajectory has a final velocity of 109 km s−1 . Assuming a distance to the heliopause of 120 AU, the spacecraft reaches interstellar space after 5.3 yr. When using an initial Sun dive to 0.6 AU instead, the solar sail obtains an escape velocity of 148 km s−1 from the solar system with a transfer time of 4.2 yr to the heliopause. Values may differ depending on the rapidity of the Sun dive and the minimum distance to the Sun. The mission concepts presented in this paper are extensions of the 0.5 kg tip mass and 196 m2 design of the successful IKAROS mission to Venus towards an interstellar solar sail mission. They allow fast flybys at Mars and into the deep solar system. For delivery (rather than fly-by) missions of a sub-kg payload the biggest obstacle remains in the deceleration upon arrival. 1. Introduction From the first medium distance ride of a steam locomo- tive in 1804, the ‘Penydarren’ of British mining engineer Richard Trevithick, at about 4 km hr−1 to the current helio- centric escape velocity of the Voyager 1 probe at 17 km s−1 , humanity has managed a velocity gain of their mechanical and automated vehicles of over four orders of magnitude within about 200 yr [1]. If technological progress continues to increase humanity’s top velocitys at this rate, that is, a doubling every 15 yr for about another seven doublings, we would reach 1 % of the velocity of light (𝑐) within little more than 100 yr from now. At this point, our travel to the stars might become reality. With our closest stellar neighbor Proxima Centauri located at a distance of about 1.3 pc (4.2 ly) [2] a velocity of 0.1 𝑐 implies a travel time of about 42 yr, well within a human lifetime and within the 46 yr of operation time of the Voyager1 mission. Recent studies suggest that the technology and knowl- edge components for interstellar missions are in reach, such as miniaturized electronic devices [3], advanced information processing [4, 5], ultra-light materials such as graphene ∗ Corresponding author (E-Mail: julius.karlapp@gmail.com). 1 Presenting author, 8th Interstellar Symposium 2023. [6] and aerographite [7] that could be used for light sails, and predictions of the light sail dynamics [8, 9, 10]. Solar sail technology has been demonstrated to work during var- ious missions in the solar system, such as IKAROS [11], NanoSail-D1, and LightSail-2 [12] (see [10, 13] for further references). An intermediate step toward interstellar travel are inter- stellar precursor missions [14], some of which have been proposed for a comet rendevous [15] and deep solar system exploration, e.g. the search for the suspected Planet Nine [10, 16]. In this paper, we explore an interstellar precursor sce- nario that we expect to be more relevant in the near future. Driven by NASA’s goal to put humans on Mars by the end of the 2030s [17] and related plans by the SpaceX company, we explore the potential of solar lightsails to carry small payloads to Mars in short times, that is, weeks to months. The energy and thus monetary demands of conventional chemical rocketry depend strongly on the Earth-Mars orbital configuration. As a result, launches are usually chosen to follow a low-energy Hohmann transfer orbit, the launch window of which opens up every 780 d or 2.1 yr. 1 https://ntrs.nasa.gov/citations/20110015650 Karlapp et al.: Paper submitted to Acta Astronautica for peer review Page 1 of 8
  • 2. Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails Nomenclature Δ𝑡 Duration Δ𝑣 Change in velocity 𝜅rad Radiation coupling constant  Absorptivity 𝜌 Aerographite density 𝜀 Shell thickness 𝑎 Acceleration 𝑎 Semi-major axis 𝑎max Maximum acceleration 𝑐 Speed of light 𝐹grav Gravitational force 𝐹rad Force of solar radiation 𝐹tot Force applied to the sail 𝑔 Earth surface gravity ℎminimum Minimum orbit height 𝑖 Inclination 𝑙 Sail radius 𝐿⊙ Solar luminosity 𝑚 Spacecraft mass 𝑀⊙ Solar mass 𝑚aero Aerographite mass 𝑀p Planets mass 𝑟 Solar distance 𝑟Hill,Earth Hill radius of Earth 𝑟Hill,Mars Hill Radius of Mars 𝑟Hill Hill radius 𝑅⊕ Radius reference body 𝑆 Sail cross sectional area 𝑡 time 𝑡deploy Time sail deployment 𝑡end Final transfer time 𝑡entry Entry time 𝑡exit,Earth Exit time from Earth’s sphere of influence 𝑡Mars Duration inside martian sphere of influence 𝑣 Velocity 𝑣Earth Orbital velocity of Earth around the Sun 𝑣end Final transfer velocity 𝑣entry Entry velocity 𝑣exit,Earth Exit velocity from Earth’s sphere of influ- ence These time scales have been proven manageable for fully robotic missions and even for more longterm missions to Mars like the Mars sample return, which are effectively a sequence of Mars missions that build on each other. Once the human factor comes into play, however, risk and emergency assessments change substantially and a wait time of months to years, e.g. for medical supply or replenishmentof essential materials or devices may be too long. In this study we feed, for the first time, results from laboratory studies on the material properties of aerographite [7] into solar sail trajectory simulations that aim at Mars and beyond. Aerographite, with its extremely low density of 0.18 kg m−3 (compared to 2700 kg m3 for aluminum foil, for example) and a radiation coupling constant 𝜅rad ∼ 1 that permits very efficient transformation of solar irradiation to acceleration, stands out as a material for solar sail ap- plications [10]. Being a carbon-based aerogel, aerographite can be described as a foam-like material with a microscopic structure composed of 𝜇m-sized carbon microtubes [18, 19, 20]. Importantfor the structural integrity of any macroscopic application, aerographite is characterized by a high elas- ticity. Elastic deformation occurs up to a compression rate of 95%. The combination of low density, efficient radiative coupling,and high elasticity makes aerographitean excellent candidate material for solar sails. Beyond solar sail concepts to Mars, we study the dynam- ical properties of aerographite sails that reach escape veloci- ties from the solar system. Hence, solar sail missions to Mars can serve as natural precursors for interstellar missions to establish the technologies for long-distance communication, deep space navigation etc. 2. Methods - Simulation Script To investigate interplanetary and interstellar transfers and to simulate possible trajectory progressions we use the Python programminglanguage in combination with the poliastro li- brary for astrodynamic calculations and visualization 2. Our objective is to use numerical calculations of the positions, velocities, and accelerations for various sail trajectories us- ing adaptive time steps. The force on our hypothetical sail is parameterized as 𝐹tot = 𝐹rad + 𝐹grav = 1 𝑟2 ( 𝐿⊙ 4𝜋𝑐 𝑆𝜅rad − 𝐺𝑀𝑚 ) , (1) where 𝑟 is the solar distance, 𝐿⊙ the solar luminosity, 𝑆 the sail cross sectional area, 𝑀 is the mass of the primary source of gravity for the sail that defines the current sphere of influence, and 𝑚 the spacecraft mass [10]. We consider an aerographite sail with 𝜅rad = 1 that is shaped as a hollow sphere with a cross-sectional area of 𝑆 = (100 m)2 = 104 m2. This corresponds to a sail radius of 𝑙 = √ 𝑆∕𝜋 = 56.4 m. We assume spherically symmetric distribution of the sail mass with a total mass of 𝑚 = 1 kg. Although our equations are ignorant as to the contribution 2 https://docs.poliastro.space/en/stable/ Karlapp et al.: Paper submitted to Acta Astronautica for peer review Page 2 of 8
  • 3. Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails of the aerographite and the payload to the total mass of the spacecraft, we can use Eq. (11) from Heller et al. [10] to estimate the aerographite mass 𝑚aero = 𝜋𝑙24𝜌𝜀, where 𝜀 is the shell thickness. For a nominal 0.1 mm thin hollow sphere with a radius of 56.4 m this gives 𝑚aero = 720 g, leaving 280 g for payload. Using a hollow aerographite hemisphere would halve the aerographite mass and permit 640 g of payload. During interplanetary transfer we assume a two-body problem of the sail and its dominant gravitational counter- part, the latter of which can be the Sun, the Earth or Mars in our case. As long as the sail is beyond the sphere of influence (SOI) of a planet, we apply Eq. (1) with 𝑀 = 𝑀⊙, or else we substitute the solar mass with the mass of the respective planet. We estimate the planetary SOI via the Hill radius 𝑟Hill = 𝑎 ⋅ 3 √ 𝑀p 3𝑀⊙ , (2) inside of which the gravitational attraction of the planet on a point mass (i.e. the sail) dominates over the gravitational force from the Sun. The Hill radius is defined by the solar mass, the mass of the planet 𝑀p, and the semi-major axis (𝑎) of the Sun-planet orbit, which is approximated to be circular. The Earth’s Hill radius is 𝑟Hill,Earth ≈ 1.5 × 106 km, while the martian Hill radius is 𝑟Hill,Mars ≈ 1.1 ×106 km [21, 22]. 3. Results In Table 1 we summarize our setup of the possible launch scenarios. We consider both direct outward transfers and inward transfers. The outward injection calculates for a launch from a parking orbit around Earth radially away from the Sun (initially defined in the gravitational sphere of influence of Earth), while the inward injection calculates for an initial conventional launch towards the Sun with a delayed sail deployment in a lower solar orbit (Sun diver approach, initially defined in the solar gravitational sphere of influence). 3.1. Outward Trajectories We now investigate the ability of our hypothetical space- craft to reach Mars or interstellar space by calculating the spacecraft position, velocity, and acceleration as a function of time. 3.1.1. Outward Mars Transfer For a direct outwards transfer from Earth to Mars, Mars needs to be near opposition since the vector of the solar radiation pressure will push the spacecraft radially away from the Sun. Thereforetwo possible starting orbits (parking orbits) around Earth are conceivable, a High Elliptical Orbit (HEO) or a Polar Orbit (PO) [23]. Previous investigations showed that other standard orbits tend to produce collisions with Earth or erratic behavior of the spacecraft [10, Sect. 3 therein]. Table 1 General orbit configurations for heliocentric injections of a solar sail spacecraft (𝑚 = 1 kg, 𝐴 = 104 m2 ). All outward injections are referring to the geocentric reference system (GCRS) because the parking orbit is defined around Earth, whilst the inward injections are defined in the heliocentric reference system (HCRS) because the spacecraft is simulated directly in the solar gravitational sphere. Heliocentric Injection Outward (GCRS) Inward (HCRS) Destination Mars Interstellar Mars Interstellar Orbit Polar Polar Solar Solar semi-major axis [𝑘𝑚] 42000 42000 149.6 ⋅ 106 149.6 ⋅ 106 Inclination [◦ ] 113.4 113.4 0 0 Eccentricity [−] 0.24 0.24 0.0167 0.0167 For an interplanetary transfer mission the use of a polar orbit is preferable (Table 1). Due to the orientation of the polar orbit the orbital plane of the spacecraft is oriented perpendicular to the solar radiation vector as shown in Fig. 1. Therefore every point along the orbit experiences a continuous irradiation from the sun (the Earth’s shadow does not matter). A time independent deployment of the sail is possible. Referring to the geocentric reference system the inclination angle of the orbit plane is 113.4◦ due to the tilt of the Earth’s spin axis, whereas in the heliocentric reference system the inclination is 90◦. The semi-major axis of the PO is chosen to be 42000 km. For smaller values the produced trajectories wont overcome the gravitational pull of the Earth. For larger values the gain in net acceleration is marginal. Since the altitude of the PO around the Earth is negligible compared to the scale of an astronomical unit (PO altitude is 2.4 × 10−4 AU) the magnitude of the solar radiation pressure can be assumed to be constant. Figure 1 shows the PO (black) around Earth (blue), chosen to calculate the outward injections following Table 1. The escaping trajectory (light blue) can be seen to align with the vector of the solar radiation pressure (red arrow) over long distances. Upon deployment of the sail (e.g. unfolding in space) in the PO, the trajectories initially spiral away from Earth due to the two acting forces of the solar radiation pressure and gravitational force of Earth. As the sail picks up speed, its trajectory resembles more and more a straight line. Any live correction of the trajectory would need additional attitude control mechanism, which we neglect in our simulations. Instead, we consider the sail a projectile under the combined photogravitational effects. The sail initially follows a curly trajectory as shown in Fig. 1. The Earth’s gravitational pull reduces the departing velocity by about 1.5 km s−1 over the course of about 1 d (Fig. 2). Thereafter, the force from the solar irradiation takes over and induces a net acceleration of the sail. At the time the sail enters interplanetary space after 3.54 d the spacecraft has an exit velocity from the Earth’s SOI of 11.2 km s−1 . 3 After the transition from the SOI of Earth to interplan- etary space, that is, into the Sun’s SOI the spacecraft enters 3The identity with the escape velocity from Earth is pure coincidence. Karlapp et al.: Paper submitted to Acta Astronautica for peer review Page 3 of 8
  • 4. Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails Earth SOI Outward Trajectory Earth Vector of Solar Radiation Pressure Parking Orbit Trajectory −8 0 8 16 24 32 Distance x [ R⊕ ] −8 0 8 16 24 32 Distance y [ R ⊕ ] −8 0 8 16 24 32 40 Distance z [ R ⊕ ] Figure 1: The polar orbit (PO, black) with the escaping trajectory (light blue) for a direct outward ejection. The PO is in the 𝑦-𝑧 plane. The solar illumination (red arrow) is along the 𝑥 axis. 0 2 4 6 8 10 12 0 0.5 1 1.5 2 2.5 3 3.5 4 transition to solar sphere of inf luence Velocity [ km s −1 ] Time [d] Outward Transfer Figure 2: Diagram of the spacecraft velocity 𝑣 as a function of time 𝑡 inside Earth’s SOI for an outward transfer. the martian SOI. Inside the martian SOI several trajectories are presented (Fig. 3). Those trajectories only differ slightly in terms of their starting point from the initial parking orbit around Earth. For the purpose of a better visual inspection these trajectories are shown in the vicinity of Mars. Any trajectory that reaches an altitude of 200 km or less above the martian surface is considered a collision (light blue) while the other trajectories (grey) escape the martian SOI. The spacecraft enters the martian SOI 613.8hr or 25.6 d after launch from the Earth’s PO (Fig. 4). After just another 4.5 hr inside the martian SOI, the spacecraft reaches Mars. During the flight inside the martian SOI the spacecraft −10 −5 0 5 10 Distance x [ R⊕ ] −10 −5 0 5 10 Distance y [ R ⊕ ] −8 −6 −4 −2 0 2 4 6 8 Distance z [ R ⊕ ] Mars SOI Trajectory Scatter Mars FlyBy Trajectory Scatter Collision Trajectory Scatter Figure 3: Three-dimensional view of several trajectories inside the martian SOI. Trajectories in light blue color collide with Mars, trajectories in grey color denote fly-bys. 65.3 65.4 65.5 65.6 65.7 65.8 65.9 613 614 615 616 617 618 619 transition to martian sphere of inf luence reaching martian orbit Velocity [ km s −1 ] Time [hr] Outward Transfer Figure 4: Diagram of the spacecraft velocity 𝑣 as a function of time 𝑡 inside the martian SOI for an outward transfer. is subject to an acceleration by the martian gravity and therefore gains 0.51 km s−1 which gives it a final velocity of 65.9 km s−1 . 3.1.2. Outward Interstellar Transfer Regarding the outward interstellar transfer we used the same polar parking orbit around Earth to deploy the spacecraft (Table 1). Instead of targeting Mars, however, all trajectories have been extended until they reached the boundary of the solar system, which we define at the distance of the heliopause at 𝑟 = 120 AU. The initial acceleration in the Earth’s orbit corresponds to the maximum acceleration of 0.33 m s−2 (dashed line in Fig. 5), corresponding to about 3 %𝑔. The acceleration then decreases following the inverse square law as a function of distance from the Sun. The factor applies also for gravita- tional forces and was added to the general thrust equation (Eq. 1). Karlapp et al.: Paper submitted to Acta Astronautica for peer review Page 4 of 8
  • 5. Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.01 0.1 1 10 100 1000 sail deployment sail deployment Earth ′ s sphere of inf luence solar sphere of inf luence Acceleration [ m s −2 ] Time [d] Outward Transfer Inward Transfer Figure 5: Diagram of the spacecraft acceleration 𝑎 as a function of time 𝑡 for both outward and inward transfers. 0 20 40 60 80 100 120 140 160 0.01 0.1 1 outward transfer time to interstellar inward transfer time to interstellar Velocity [ km s −1 ] Time [ yr ] Outward Transfer Inward Transfer Figure 6: Diagram of the spacecraft’s velocity 𝑣 as a function of time 𝑡 in interplanetary space. As a result, the velocity of the spacecraft increases continuously, but ever less with increasing distance to the Sun. The spacecraft ultimately reaches a terminal velocity of 𝑣end = 109 km s−1 after a duration of about 2 yr (Fig. 6). At this maximum speed, the spacecraft reaches the boundary of the solar system after 𝑡end = 5.3 yr. We executed a limited series of trajectory simulations with variations of the launch configurations from the initial parking orbit around the Earth and found none that yielded any significant increase of the terminal velocity. The ter- minal velocity of the spacecraft can only be increased by reducing its overall mass or by increasing its cross sectional surface. 3.2. Inward Trajectories An alternative approach to increase the terminal velocity of the spacecraft is to reduce the distance to the Sun for the initial sail deployment. The increased photon pressure in −2 −1.5 −1 −0.5 0 0.5 1 Distance x [AU] −2 −1.5 −1 −0.5 0 0.5 1 D istance y [AU ] −1.5 −1 −0.5 0 0.5 1 1.5 Distance z [AU] Possible Directions of Interstellar Trajectories Sun Earth Δvbreaking = 4.5 km s−1 6.0 7.6 9.1 10.6 12.1 13.6 15.1 16.6 Trajectory Scatter Earth Orbit Minimum Distance Figure 7: The Sun diver approach allows the spacecraft to achieve several radial trajectory directions inside Earth’s orbital plane around the Sun. The required breaking maneuver in the parking orbit around Earth is labeled Δ𝑣breaking for each direction. Steeper maneuvers require a higher amount of Δ𝑣. the vicinity to the Sun results in a higher radiation force component and therefore a shorter transfer time than the direct outward approach from Earth. To prove this thesis our script was modified to simulate a two phase flight. For the first phase a conventional impulse maneuver using chemical engines in an orbit around Earth towards the Sun is executed, lowering the spacecraft’s orbit around the Sun (Table 1). Once it reaches a predefined solar distance of 0.6 AU (black circle in Fig. 7), we assume that the sail is deployed, which initiates the second flight phase. From that point on, the spacecraft accelerates radially out- ward and therefore is pushed away from the Sun. We tested this method both for a transfer to Mars and to interstellar trajectories. In addition to an increase of the terminal velocity this Sun diver approach offers another advantage. By adjusting the breaking velocity from the parking orbit around Earth the rapidity of the fall towards the Sun can be controlled. In Fig. 7 we present the trajectory scatter for various val- ues of Δ𝑣breaking. The Δ𝑣 was calculated considering the spacecrafts standard configuration with respect to the initial velocity of Earth’s orbit (𝑣Earth = 29.78 km s−1 ). Steeper dives require stronger breaking. By applying the correct amount of Δ𝑣breaking or deploying the sail at the second intersection between the initial trajectory and the minimum distance every direction is possible inside Earth’s orbital plane. To implementthe Sun divermaneuversome assumptions had to be made. To minimize the required Δ𝑣 for the break- ing maneuverat Earth,the minimumdistance aroundthe Sun (the deploymentaltitude) must not be too low. Yet, in orderto reduce the transfer time and to exploit the advantages of the first-in-then-out approach the minimum distance needs to be sufficiently low. Furthermore, for a single impulse maneuver in Earth’s SOI, Mars needs to be located in the leading Karlapp et al.: Paper submitted to Acta Astronautica for peer review Page 5 of 8
  • 6. Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Distance x [AU] −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 D istance y [AU ] −1 −0.5 0 0.5 1 1.5 Distance z [AU] Inward Mars Transfer Sun Earth Mars Position Earth on Arrival Position Mars at Launch Trajectory Earth Orbit Mars Orbit Minimum Distance Figure 8: Solar system showing Earth’s and Mars’ orbit around the Sun. The calculated trajectory for the inward Mars transfer reaches Mars in the leading node of both orbital planes. node of the two orbital planes upon arrival of the space- craft. Multi-impulse maneuvers are also possible but have not been tested for this publication. With a multi-impulse maneuver the inclination change can be compensated and furthermorethe range of possible trajectories around the Sun be increased. 3.2.1. Inward Mars Transfer For the Mars transfer a random constellation between Earth and Mars was generated. As a boundary condition, however, we searched for trajectories from Earth that lead to encoun- ters with Mars when Mars is in the ascending node of the two orbital planes. In these cases the spacecraft does not need to leave the ecliptic and we can consider the problem in a plane. To inject the spacecraft towards the Sun, a negative impulse of Δ𝑣 ≈ 4.5 km s−1 needs to be applied in the parking orbit around Earth. The minimum solar distance for sail deployment was again set to 0.6 AU. Our numerical script contains two loops referring to the solar SOI as well as the martian SOI. In Fig. 8 the solar system is shown with the additional orbit of Mars (orange) next to the orbit of Earth (blue). Both the position of Earth on arrival and the position of Mars at launch are shown faded. The required trajectory to reach Mars is presented in green. Using the standard spacecraft configuration we calcu- late a total transfer time of 125.6 d or 4.2 months (Fig. 9). When reaching the martian orbit the spacecraft has a ve- locity of 118.02 km s−1 . Considering the entry velocity of ≈ 117.85 km s−1 , we find that it gained Δ𝑣 = 0.17 km s−1 inside the martian SOI. Due to the high velocity the time spent in the martian SOI is just Δ𝑡Mars ≈ 2.5 hr. In summary, the sail was deployed after it reached the minimum solar distance of 0.6 AU about 103 d after launch. The maximum acceleration was determined as roughly 0.11 m s−2. 117.8 117.85 117.9 117.95 118 118.05 3012.5 3013 3013.5 3014 3014.5 3015 3015.5 Velocity [ km s −1 ] Time [hr] Inward Transfer transistion to martian sphere of inf luence reaching martian orbit Figure 9: Diagram of the spacecraft’s velocity 𝑣 as a function of time 𝑡 inside the martian SOI for an inward transfer. −1 −0.5 0 0.5 1 Distance x [AU] −1 −0.5 0 0.5 1 D istance y [AU ] −1 −0.5 0 0.5 1 Distance z [AU] Inward Interstellar Trajectory Sun Earth Trajectory Earth Orbit Minimum Distance Figure 10: Solar system showing Earth’s orbit around the Sun. The calculated trajectory for the inward interstellar transfer escapes the solar system in a straight line. 3.2.2. Inward Interstellar Transfer For the inward interstellar transfer the same approach was used as in subsection 3.2.1 for the Sun diver maneuver to Mars (Table 1). Starting in a solar orbit equally to the Earth’s orbit the orbit of the spacecraft is lowered by initiating a neg- ative impulse maneuver. By reaching the minimum_distance the sail is deployed and the solar radiation pressure pushes the spacecraft away from the Sun. Yet, instead of stopping the calculating function when reaching the Mars SOI the tra- jectories are continued. As in subsection 3.1.2 the function then is stopped when the spacecraft reaches a distance of 𝑟 = 120 AU. Again the trajectories showed a linear progres- sion after performing the Sun diver maneuver in the close vicinity of the Sun (Fig. 10). As shown in Fig. 5 a spacecraft with standard con- figuration on an inward interstellar transfer experiences a maximum acceleration of 0.11 m s−2 occurring after 103 d. Karlapp et al.: Paper submitted to Acta Astronautica for peer review Page 6 of 8
  • 7. Ultrafast transfer of low-mass payloads to Mars and beyond using aerographite solar sails After the sail deployment the velocity increases again to its maximum of 148 km s−1 (Fig. 6). This maximum velocity is reached after about 300 d. Flying with its maximum velocity the spacecraft reaches the boundary of the solar system after roughly 1520 d or 𝑡end = 4.16 yr. Even closer solar approaches than those at 0.6 AU that we tested are certainly possible. For context, the still active Parker Solar Probe has previously survived several hours interior to 0.1 AU and as close as 0.086 AU from the Sun [24]. 4. Conclusions We find that a 1 kg solar sail with a cross-sectional area of 𝑆 = (100 m)2 = 104 m2 (radius of 56 m) can be accelerated to a velocity of 𝑣entry = 65 km s−1 relative to Mars in a direct outward transfer from Earth (when Mars is near opposition). In this scenario, Mars is reached 26 d after launch. An initial Sun diver maneuver down to 0.6 AU from the Sun increases the travel time and flyby velocities to Mars but allows launches from various Earth-Mars orbital constellations. Our simulations of direct launches to an interstellar tra- jectory reveal terminalescape velocities of 𝑣end = 109 km s−1 and passage through the heliopause after 5.3 yr. If instead the sail is first directed to the Sun, then a final velocity of 𝑣end = 148 km s−1 is obtained with a transfer time of 𝑡end = 4.2 yr to the heliopause. Details depend on how fast the sail is directed to the Sun (e.g. using conventional chemical vehicles) and how close its perihelion distance. The trajectories and solar sail properties presented in this paper can be considered as a next step after the successful IKAROS mission to Venus, which had a comparable mass (0.5 kg) but smaller sail area 196 m2. The key difference is in our assumption of aerographite as a sail material compared to the polyimide sheet of IKAROS, which had a mass of about 0.01 kg m−2 compared to 7.6 × 10−7 kg m−2 for aerographite. An as of yet unresolved question about our concept is the decelerationof the payloadif this is supposed to be a delivery and not a flyby mission. One possible scenario that we suggest for future studies is the deceleration using friction with the Martian atmosphere. Once dropped on the martian surface, radioactive contamination of the payload can help to recovereven gram-sizedpayloadon the Martian surface.The key challenges would, of course, be the structural integrity of the payload (not necessarily of the sail) during descent given the expected high acceleration and temperature. 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