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Exploring Equations

Introduction

Interstellar exploration requires many different skills sets and areas of knowledge. Those not comfortable with mathematics equations may find other areas of this website more relevant to their interests. However, for those who are familiar with basic mathematics, much can be learned from an understanding of some key equations that underpin the physics of interstellar propulsion. The material below is intended to help you explore these equations, by allowing you to investigate graphically how changes in important variables alter the result of the equation.

Initially, this page includes two equations: The Tsiolkovsky rocket equation, and Robert L. Forward’s equation for the acceleration of a light sail. We plan to expand the page with more equations in future. If you have any suggestions for the equations you’d like to see here, please email us at equations@i4is.org, and we’ll do our best to add the most popular suggestions.

The Tsiolkovsky rocket equation

\[ \Delta v = v_e \ln \left(\dfrac{m_0}{m_f}\right) \]

This equation gives the velocity increase available from a single-stage rocket with initial mass \(m_0\) (payload, structure and fuel), final mass \(m_f\) (only payload and structure, as no fuel left) and rocket exhaust velocity \(v_e\).

How it works

The Tsiolkovsky rocket equation is one of the most important equations across the whole topic of spaceflight. It applies to any single-stage rocket which propels itself forwards by expelling material out of the back of the rocket. It works for any rocket, from the smallest satellite thruster to the huge engines used for the Apollo missions to the moon, and even the fusion rockets proposed for the Daedalus and Icarus interstellar probes.

For a given choice of propellants, the exhaust velocity will be a broadly fixed value, as it depends on the physics or chemistry underlying the propulsion approach. For example, when the fuel is liquid hydrogen and the oxidiser is liquid oxygen (as used in the Space Shuttle Main Engine), the rocket exhaust velocity is ve = 4,500 metres per second (approximately). The key feature of this equation is that the answer – the velocity increase in the rocket between ignition and when all the propellant is used up, known as the ‘delta-V’ and shown in the equation as ΔV, using the Greek letter delta – depends on the logarithm of the ‘mass ratio’ (m0/mf). Those familiar with the logarithmic function (or its inverse, the exponential function) will recognise that delta-V will only increase slowly with the mass ratio.

The result is that, for any practically achievable rocket, where the mass ratio is typically between about ten (Saturn 1B) and forty (Ariane 5), the delta-V achievable will be between two and four times the rocket’s exhaust velocity. This is why the chemical propellants typically used in today’s rockets, to launch payloads to Earth orbit or within the solar system, won’t be of any use for interstellar journeys: because the delta-V we need, if we’re to get our payload to an interstellar destination in a reasonable timeframe (e.g. below a century) is much more than four times the exhaust velocity of any chemical propellants available to us.

That’s why those interested in interstellar propulsion focus on rockets using nuclear fission, nuclear fusion or even more advanced technologies, each of which promises much higher exhaust velocities than available from chemical rockets. However, the Tsiolkovsky equation still applies to them too!

Historical background

The equation was developed and published in 1903 by the Russian rocket pioneer Konstantin Tsiolkovsky (1857-1935). If you’d like to know more about this visionary genius, we published a biographical article on him by John I Davies in Principium issue 20 (February 2018), available elsewhere on this website.

Explore!

See how the \(\Delta v\) depends on the initial mass \(m_0\) for a specific rocket exhaust velocity \(v_e\) (with units m/s) and final mass \(m_f\) (with units tonnes \( = 1000\) kg).

\( \Delta v = \)\( \ln (m_0/ \)\( ) \)

The mass ratio (\(=m_0/m_f\)) is plotted below for an exhaust velocity of \(v_e=1000\) m/s.

Light Sail Acceleration

Robert L. Forward provides an equation [1] to calculate the acceleration of a light sail given the mass of the spacecraft, reflectance of the light sail, and power of the laser.

\[ \alpha = \dfrac{2 \eta P}{Mc} \]

where \(\alpha\) is the acceleration of the spacecraft, \(\eta\) is the reflectance of the light sail, \(P\) is the incidence laser power, and \(M\) is the total mass of the spacecraft including payload, structure and sail. \(c\) is the speed of light.

How it works

A light sail is a propulsion technology that uses the momentum carried by photons of light to accelerate a large, thin and very light sheet of reflective material, plus its payload. The source of these light photons may be natural (a ‘solar sail’) or artificial (a ‘laser sail’). The key benefit of using a light sail, rather than a rocket, is that it does not need to carry its own propellant, so is not subject to the limitations encoded in the Tsiolkovsky rocket equation above. The main disadvantage of this form of propulsion is that the momentum carried by a photon of light is very small, so a successful light sail mission will require a combination of a strong source of light, a large sail area, and a light spacecraft (including both the mass of the sail and that of the payload).

Solar sails are already a practical technology: the Japan Space Agency (JAXA) launched IKAROS (Interplanetary Kite-craft Accelerated by Radiation Of the Sun) towards Venus in 2010, and received signals from it for five years, until 2015, while the Planetary Society launched their LightSail 2 spacecraft into Earth orbit in 2019, and it was still operating at the end of 2021, some 2.5 years after launch.

Laser sails, on the other hand, have not yet been demonstrated in space – but the i4is Glowworm project, and its target probe ‘Pinpoint’, aim to pioneer this. In the longer term, the Breakthrough Starshot team have the aim of launching small laser sailcraft to our nearest star beyond the Sun, Proxima Centauri, at 20% of the speed of light, using a giant laser.

Historical background

Solar sails have frequently featured in science fiction, with one of the earliest examples being Arthur C. Clarke’s 1964 short story ‘Sunjammer’ (later renamed ‘The Wind from the Sun’). In the 1970s, Dr Louis Friedman of the Jet Propulsion Laboratory (JPL) proposed to NASA a solar sail-powered spacecraft to rendezvous with the 1986 return of Halley’s Comet, but the project was cancelled after a year’s design work. Interest in the practical possibilities of light sails grew after that, leading to Robert L. Forward’s 1984 paper on the potential use of laser sails in interstellar travel, from which the equation discussed on this page is taken (See ‘Roundtrip Interstellar Travel Using Laser-Pushed Lightsails’).

Explore!

See how the acceleration \(\alpha\) depends on the properties of the spacecraft, light sail and incident light. Reflectance \(\eta\) is the percentage of incident light reflected (expressed as a decimal value between 0 and 1). Incident laser power \(P\) has units GW, and total spacecraft mass \(M\) has units kg. The units of acceleration are shown as m/s2.

\( \alpha = ( 2 \times \) \( \times \)\( ) / ( M c ) \)

Challenge!

You are designing a light sail mission! You want the acceleration \(\alpha\) of the light sail to be as high as possible.

  • Which of the parameters \(\eta\), \(P\) and \(M\) have an effect on the acceleration \(\alpha\) of the light sail?
  • When designing the mission, would you want to make the spacecraft as heavy as possible?
  • Would you choose a material for the sail with a high reflectance or a low reflectance?
  • Would you want a laser with a high power or a low power?
Answers

All of the three parameters have an effect on the acceleration. As the total mass of the spacecraft increases the acceleration decreases, therefore you would want the spacecraft mass to be as small as possible to maximise your acceleration. You would want the light sail to reflect as much light as possible. This is also important to prevent the light overheating the spacecraft. You would want the incident power on the sail to be as high as possible.

References

  1. Forward RL. Roundtrip interstellar travel using laser-pushed lightsails. Journal of Spacecraft and Rockets [Internet]. 1984 Mar;21(2):187–95. Available from: https://dx.doi.org/10.2514/3.8632 and https://pdfs.semanticscholar.org/25b2/b991317510116fca1e642b3f364338c7983a.pdf