According to very rough estimates, the monarch butterfly:

- can fly around 100,000-600,000 m/kJ
- and move mass at around 0.065-0.36 kg.m/J

The **Monarch Butterfly **is a butterfly known for its impressively long migration.^{1}

The average mass of a monarch butterfly prior to its annual migration has been estimated to be 600mg^{2}

The following table gives some very rough estimates of energy expenditures, speeds and distances for several modes of flight, based on confusing information from a small number of papers (see footnotes for details).

Activity | Description | Energy expenditure per mass ( J/g.h) | Energy expenditure for 600mg butterfly (J/s) | Speed (m/s) | distance/energy (m/J) |

Soaring/gliding | Unpowered flight, including gradual decline and ascent via air currents | 8-33^{3} | ~0.0014 – 0.0056^{4} | Very roughly 2.5-3.6 on average^{5} | 446- 2571^{6} |

Cruising | Low speed powered flight | Very roughly 209^{7} | 0.035^{8} | Maximum: >5^{9} | Maximum: >143^{10} |

Sustained flapping | High speed powered flight | Very roughly 837^{11} | ~0.14^{12} | Maximum: >13.9^{13} | Maximum: >99^{14} |

Soaring is estimated to be potentially very energy efficient (see Table 1), since it mostly makes use of air currents for energy. It seems likely that at least a small amount of powered flight is needed for getting into the air, however monarch butterflies can apparently fly for hundreds of kilometers in a day^{15}, so supposing that they don’t stop many times in a day, taking off seems likely a negligible part of the flight.^{16}

This would require ideal wind conditions, and our impression is that in practice, butterflies do not often fly very long distances without using at least a small amount of powered flight.^{17}

There is stronger evidence that monarch butterflies can realistically soar around 85% of the time, from Gibo & Pallett, who report their observations of butterflies under relatively good conditions.^{18} So as a high estimate, we use this fraction of the time for soaring, and suppose that the remaining time is the relatively energy-efficient cruising, and take the optimistic end of all ranges. This gives us:

One second of flight = 0.15 seconds cruising + 0.85 seconds soaring

= 0.15s * 5 m/s cruising + 0.85s * 3.6m/s soaring

= 0.75m cruising + 3.06m soaring = 3.81m total

= 0.75m / 143 m/J cruising + 3.06m / 2571 m/J soaring = 0.0064 J total

Thus distance/energy = 3.81m/0.0064 J = 595 m/J

For a low estimate of efficiency, we will assume that all of the powered flight is the most energetic flight, that powered flight is required half the time on average, and that the energy cost of gliding is twice that of resting. This gives us:

Energy efficiency = (50% * soaring distance + 50% * powered distance)/(50% * soaring energy + 50% * powered energy)

= (50% * soaring distance/time + 50% * powered distance/time)/(50% * soaring energy/time + 50% * powered energy/time)

= (0.5 * 2.5m/s + 0.5 * 13.9m/s)/(0.5 * (0.0056 * 2) J/s + 0.5 * 0.14 J/s)

= 108 m/J

Thus we have, very roughly:

distance/energy = 100,000-600,000 m/kJ

As noted earlier, the average mass of a monarch butterfly prior to its annual migration has been estimated to be 600mg^{19}

Thus we have:

mass*distance/energy = 0.0006 kg * 108 — 0.0006 kg * 595 m/J

= 0.065 — 0.36 kg.m/J

The wandering albatross:

- can fly around 240m/kJ
- and move mass at around 1.4—3.0kg.m/J

The wandering albatross is a very large seabird that flies long distances on wings with the largest span of any bird.^{1}

In a study of wandering albatrosses flying in various wind speeds and directions, average ground speed was 12 m/s, though the fastest ground speed measured appears to be around 24m/s, ^{2} We use average ground speed for this estimate because we only have data on average energy expenditure, though it is likely that higher ground speeds involve more energy efficient flight, since albatross flight speed is dependent on wind and it appears that higher speeds are substantially due to favorable winds.^{3}

One study produced an estimate that when flying, albatrosses use 2.35 times their basal metabolic rate^{4} which same paper implies is around 1,833 kJ/bird.day.^{5}

That gives us a flight cost for flying of 0.050 kJ/second.^{6}

This gives us a distance per energy score of:

distance/energy

= 12 m/s / 0.050 kJ/s

= 240m/kJ

Albatrosses weigh 5.9 to 12.7 kg.^{7}

Thus we can estimate:

mass.distance/Joule

= 5.9kg * 240 m/kJ to 12.7kg*240 m/kJ

= 1.4—3.0kg.m/J

We estimate that a record-breaking two-person paramotor:

- covered around = 0.36 m/kJ
- and moved mass at around 0.058 – 0.10 kg.m/J

Paramotors are powered parachutes that allow the operator to steer.^{1}

The Fédération Aéronautique Internationale (FAI) maintains records for a number of classes of paramotor contest. We look at subclass RPF2T—(Paramotors : Paraglider Control / Foot-launched / Flown with two persons / Thermal Engine)—which is appears to be the most recent paramotor record for ‘Distance in a straight line with limited fuel’.^{2}

The record distance was 123.18 km.^{3} The FAI rules state that no more than 7.5 kg of fuel may be used.^{4} We will assume that in the process of breaking this record, all of the available fuel was used. We will also assume that regular gasoline was used. Gasoline has an energy density of 45 MJ/kg.^{5}

Distance per energy = 123.18 km / (7.5 kg * 45 MJ/kg)

= 0.36 m/kJ

The weight of an entire paramotoring apparatus appears to be the weights of the passengers plus motor plus wing plus clothing and incidentals, based on forum posts.^{6} These posts put clothing and incidentals at around 8kg, but are estimates for single person flying, whereas this record was a two person flight. We guess that two people need around 1.5x as much additional weight, for 12kg.

Wikipedia says that the weight of a paramotor varies from 18kg to 34 kg.^{7} However it is unclear whether this means the motor itself, or all of the equipment involved.

The glider used appears to be MagMax brand, a typical example of which weighs around 8kg, though this may have been different in 2013, or they may have used a different specific glider.^{8} To account for this uncertainty, we shall add the glider weight to the high estimate, and so estimate the weight of the glider and motor together at 18-42kg.

We will assume that the apparently male pilots weighed between 65 and 115 kgs each, based on normal male weights^{9}.

Thus we have:

weight = motor + wing + people + clothing and incidentals

weight (low estimate) = 18 + 65*2 + 12 = 160kg

weight (high estimate) = 42 + 115*2 + 12 = 284kg

High efficiency estimate:

284kg * 0.36 m/kJ = 0.10 kg.m/J

Low efficiency estimate:

160kg * 0.36 m/kJ = .058 kg.m/J

This gives us a range of 0.058 – 0.10 kg.m/J

The Spirit of Butt's Farm:

- covered around 31.67 m/kJ
- and moved mass at around 0.16 – 0.086 kg.m/J

The Spirit of Butt’s Farm:

- covered around 31.67 m/kJ
- and moved mass at around 0.16 – 0.086 kg.m/J

The Spirit of Butt’s Farm was a record setting model airplane that crossed the Atlantic on one gallon of fuel.^{1} Fully fueled it weighed 4.987 kg, dry it weighed 2.705 kg.^{2}

The record setting flight used 117.1 fluid ounces of fuel.^{3} The straight line distance of the flight was 3,028.1 km.^{4} It was powered by 88% Coleman lantern fuel, mixed with lubricant.^{5} Coleman fuel is based on naphtha ^{6}, so we can use the energy density of naphtha—31.4 MJ/L^{7}—as a rough guide to its energy content, though naphtha appears to vary in its content, and it is unclear whether Coleman fuel consists entirely of naphtha.

From all this, we have:

Distance per energy = 3,028.1 km / (117.1 fl oz * 0.88 * 31.4 MJ/L)

= 31.67 m/kJ

For weight times distance per energy we will calculate a best and a worst score. To calculate the best score we will use the fully fueled weight, and to calculate the worst score we will use the dry weight. All other values are the same in both calculations.

Best score:

Distance*mass/energy = 4.987 kg * 31.67 m/kJ

= 0.16 kg.m/J

(4.987 kg * 3,028.1km) / (117.1 US fluid ounces * 31.4MJ/litre) = 0.1389 kg*m/j

Worst score:

Distance*mass/energy = 2.705 kg * 31.67 m/kJ

= .086 kg.m/J

Photo by Ronan Coyne, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license, unaltered.

]]>The MacCready *Gossamer Albatross*:

- covered around 2.0—4.6 m/kJ
- and moved mass at around 0.1882 —0.4577 kg.m/J

The **MacCready Gossamer Albatross **was a human-powered flying machine that crossed the English Channel in 1979.

We do not know the pilot’s average power output, however:

- Wikipedia claims at least 300W was required to fly the craft
^{4} - Chung 2006, an engineering textbook, claims that the driver, a cyclist, could produce around 200W of power.
^{5} - Our impression is that 200W is a common power output over houres for amateur cycling. For instance, one of our researchers is able to achieve this for three hours.
^{6}

The best documented human cycling wattage that we could easily find is from professional rider Giulio Ciccone who won a stage of the Tour de France, then uploaded power data to the fitness tracking site Strava.^{7} His performance suggests around 318W is a reasonable upper bound, supposing that the pilot of the *Gossamer Albatross* would have had lower performance.^{8}

To find the energy used by the cyclist, we divided power output by typical efficiency for a human on a bicycle, which according to Wikipedia ranges from .18 to .26.^{9}

For distance per energy this gives us a highest measure of:

35.7 km / ((200W * (2 hours + 49 minutes))/0.26) = 4,577 m/MJ

And a lowest measure of:

35.7 km / ((318W * (2 hours + 49 minutes))/0.18) = 1,993 m/MJ

For weight times distance per energy this gives us a highest measure of:

(100kg * 35.7 km) / ((200W * (2 hours + 49 minutes))/0.26) = 0.4577 kg*m/j

And a lowest measure of:

(100kg * 35.7 km) / ((318W * (2 hours + 49 minutes))/0.17) = 0.1882 kg*m/j

The Boeing 747-400:

- covers around 0.0021m/kJ.
- and moves mass at around 0.39 – 0.83 kg.m/J

The *Boeing 747-400 *is a 1987 passenger plane.^{1}

The plane uses 10.77 kg/km of fuel on a medium haul flight.^{2} We do not know what type of fuel it uses, but typical values for aviation fuel are around 44MJ/kg.^{3} Thus to fly a kilometer, the plane needs 10.77 kg of fuel, which is 10.77 x 44 MJ = 474 MJ of fuel. This gives us 0.0021m/kJ.

According to Wikipedia, the 747’s ‘operating empty weight’ is 183,523 kg and its ‘maximum take-off weight’ is 396,893 kg.^{4} We use the range 183,523 kg—396,893 kg since we do not know at what weight in that range the relevant speeds were measured.

We have:

- Distance per kilojoule: 0.0021m/kJ
- Mass: 183,523 kg—396,893 kg

This gives us a range of 0.39 – 0.83 kg.m/J

The Airbus A320:

- covers around 0.0078 m/kJ
- and moves mass at around 0.33 – 0.61 kg.m/J

The *Airbus A320 *is a 1987 passenger plane.^{1}

The plane uses 2.91 kg of fuel per km on a medium haul flight.^{2} We do not know what type of fuel it uses, but typical values for aviation fuel are around 44MJ/kg.^{3} Thus to fly a kilometer, the plane needs 2.91kg of fuel, which is 2.91 x 44 MJ = 128MJ of fuel. This gives us 0.0078 m/kJ

According to modernairliners.com, the A320’s ‘operating empty weight’ is 42,600 kg and its ‘maximum take-off weight’ is 78,000 kg.^{4} We use the range 42,600—78,000 kg, since we do not know at what weight in that range the relevant speeds were measured.

We have:

- Distance per kilojoule: 0.0078 m/kJ
- Mass: 42,600—78,000 kg

This gives us a range of 0.33 – 0.61 kg.m/J

The North American P-51 Mustang:

- flew around 0.073—0.092 m/kJ
- and moved mass at around 0.25 – 0.50 kg.m/J

The *North American P-51 Mustang *was a 1940 US WWII fighter and fighter-bomber.^{1}

According to Wikipedia^{2}:

**Empty weight:**7,635 lb (3,465 kg)**Gross weight:**9,200 lb (4,175 kg)**Max takeoff weight:**12,100 lb (5,488 kg)

We use the range 3,465—5,488 kg, since we do not know at what weight in that range the relevant speeds were measured.

Wikipedia tells us that cruising speed was 362 mph (162 m/s)^{3}

A table from *WWII Aircraft Performance* gives combinations of flight parameters apparently for a version of the P-51, however it has no title or description, so we cannot be confident. ^{4} We extracted some data from it here. This data suggests the best combination of parameters gives a fuel economy of 6.7 miles/gallon (10.8km)

We don’t know what fuel was used, but fuel energy density seems likely to be between 31—39 MJ/L = 117—148 MJ/gallon.^{5}

Thus the plane flew about 10.8km on 117—148 MJ of fuel, for 0.073—0.092 m/kJ

We have:

- Distance per kilojoule: 0.073—0.092 m/kJ
- Mass: 3,465—5,488 kg

This gives us a range of 0.25 – 0.50 kg.m/J

The Vickers Vimy:

- flew around 0.025—0.050 m/kJ
- and moved mass at around 0.081 – 0.25 kg.m/J

The *Vickers Vimy *was a 1917 British WWI bomber.^{1} It was used in the first non-stop transatlantic flight.

According to Wikipedia^{2}:

**Empty weight:**7,104 lb (3,222 kg)**Max takeoff weight:**10,884 lb (4,937 kg)

We use the range 3,222—4,937 kg, since we do not know at what weight in that range the relevant speeds were measured.

We also have:

**Power:**360 horsepower = 270 kW^{3}**Efficiency of use of energy from fuel:**we did not find data on this, so use an estimate of 15%-30%, based on what we know about the energy efficiency of the Wright Flyer.

From these we can calculate:

Energy use per second

= power of engine x 1/efficiency in converting energy to engine power

= 270kJ/s / .15—270kJ/s / .30

= 900—1800 kJ/s

Wikipedia gives us:

100 mph (160 km/h, 87 kn)**Maximum speed:**^{4}

Note that the figures for power do not obviously correspond to the highest measured speed. This is a rough estimate.

We now have (from above):

- speed = 100 miles/h = 44.7m/s
- energy use = 900—1800 kJ/s

Thus, on average each second the plane flies 44.7 m and uses 900—1800 kJ, for 0.025—0.050 m/kJ.

We have:

- Distance per kilojoule: 0.025—0.050 m/kJ
- Mass: 3,222—4,937 kg

This gives us a range of 0.081 – 0.25 kg.m/J

The Wright model B:

- flew around 0.10—0.21m/kJ
- and moved mass at around 0.036 – 0.12 kg.m/J

The *Wright Model B* was a 1910 plane developed by the Wright Brothers.^{1}

According to Wikipedia^{2}:

**Empty weight:**800 lb (363 kg)**Gross weight:**1,250 lb (567 kg)

We use the range 363—567 kg, since we do not know at what weight in that range the relevant speeds were measured.

From Wikipedia, we have^{3}:

**Power:**35 horsepower = 26kW**Efficiency of use of energy from fuel:**we could not find data on this, so use an estimate of 15%-30%, based on what we know about the energy efficiency of the Wright Flyer.

From these we can calculate:

Energy use per second

= power of engine x 1/efficiency in converting energy to engine power

= 26kJ/s / .15—26kJ/s / .30

= 86.6—173 kJ/s

Wikipedia gives us^{4}:

**Maximum speed:**45 mph (72 km/h, 39 kn)**Cruise speed:**40 mph (64 km/h, 35 kn)

We use the cruise speed, as it seems more likely to represent speed achieved with the energy usages reported.

We now have (from above):

- speed = 40miles/h = 17.9m/s
- energy use = 86.6—173 kJ/s

Thus, on average each second the plane flies 17.9m and uses 86.6—173 kJ, for 0.10—0.21m/kJ.

We have:

- Distance per kilojoule: 0.10—0.21m/kJ
- Mass: 363—567kg

This gives us a range of 0.036 – 0.12 kg.m/J