Published 25 November, 2020; last updated 10 December, 2020

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 migration across North America.¹

The average mass of a monarch butterfly prior to its annual migration has been estimated to be 600mg²

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).

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 day¹⁵, 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

This also gives us:

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

= 0.0064 J total

Thus we have:

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

For concreteness, a kJ is the energy in around a quarter of a raspberry. ¹⁹

As noted earlier, the average mass of a monarch butterfly prior to its annual migration has been estimated to be 600mg²⁰

Thus we have:

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

= 0.065 — 0.36 kg⋅m/J

Primary author: Ronny Fernandez

1. “The monarch butterfly, Danaus plexippus, is famous for its spectacular annual migration across
   North America”

   Zhan, Shuai, Wei Zhang, Kristjan Niitepõld, Jeremy Hsu, Juan Fernández Haeger, Myron P. Zalucki, Sonia Altizer, Jacobus C. de Roode, Steven M. Reppert, and Marcus R. Kronforst. “The Genetics of Monarch Butterfly Migration and Warning Colouration.” Nature 514, no. 7522 (October 2014): 317–21. https://doi.org/10.1038/nature13812.

2. “Each year in late summer and fall in southern Ontario, monarch butterflies, Danaus plexippus plexippus L., engage in migratory flight to the southern U.S.A. and Mexico…

   …Late summer monarchs average approximately 600mg.”
   
   Gibo, David L., and Pallett, Megan J., “Soaring Flight of Monarch Butterflies, Danaus Plexippus (Lepidoptera: Danaidae), during the Late Summer Migration in Southern Ontario.” Canadian Journal of Zoology, July 1979. https://doi.org/10.1139/z79-180.

3. Gibo and Pallet (1979) appear to think energy expenditure of soaring is well approximated by energy expenditure for resting:
   
   “The most efficient flying technique, in terms of
   cost per unit of distance travelled, is soaring (Pennycuick 1969, 1975). During soaring flight, altitude
   is gained or maintained by gliding in rising air currents. Since a soaring animal is actually gliding,
   the wings are held more or less motionless and
   the high-energy expenditure of powered flight is
   avoided…
   
   …if soaring requires approximately
   the basal level of metabolic expenditure, an average D. piesippus with an initial fat supply of 140 mg,
   which could fly under power for only I1 h, may be
   able to soar for 1060 h.”
   
   Gibo, David L., and Pallett, Megan J., “Soaring Flight of Monarch Butterflies, Danaus Plexippus (Lepidoptera: Danaidae), during the Late Summer Migration in Southern Ontario.” Canadian Journal of Zoology, July 1979. https://doi.org/10.1139/z79-180.
   
   They say, “Flying Lepidoptera have a metabolic rate that is as
   much as 100 times above the basal rate (Zebe 1954).”
   
   Zhan et al 2014 measure metabolic rates for monarch butterflies specifically, but only report that the resting rate is 25x lower than a flying rate they prompted:
   
   “…we tested it by measuring flight metabolic
   rates. We found active flight to be exceptionally demanding energetically, utilizing 25 times
   more energy than resting…”
   
   Zhan, Shuai, Wei Zhang, Kristjan Niitepõld, Jeremy Hsu, Juan Fernández Haeger, Myron P. Zalucki, Sonia Altizer, Jacobus C. de Roode, Steven M. Reppert, and Marcus R. Kronforst. “The Genetics of Monarch Butterfly Migration and Warning Colouration.” Nature 514, no. 7522 (October 2014): 317–21. https://doi.org/10.1038/nature13812.
   
   Also, their data makes very little sense to us. For instance, the figure suggests that flying is less energy intensive than resting. However it seems likely that this is a misunderstanding on our part, since for instance their y-axis in not labeled. Given this and the earlier claim that the difference between resting and flying can be 100x for butterflies, we use the relative number 25-100x, in combination with the flight and cruising energy estimates given below.
   
   high estimate:
   837/25 J/g⋅hr = 33 J/g⋅hr (see below for flying rate)
   
   The lowest estimate would be the cruising speed energy expenditure divided by 100, but it seems like the 100 figure was meant as an upper bound on the difference, so probably referring to rest vs. maximal flight energy.
   
   Thus we take the lowest of the other combinations, as low end estimate:
   min(209/25, 837/100) = min(8.4, 8.4) = 8.4 J/g⋅hr
4. 8-33 J/g⋅hr * 0.6g= ~5-20 J/hr = 0.0014 – 0.0056 J/s
5. Gibo & Pallett (1979) assume this in estimating flight range possible via soaring:
   
   “Consequently, if soaring requires approximately
   the basal level of metabolic expenditure, an average D. piesippus with an initial fat supply of 140 mg,
   which could fly under power for only I1 h, may be
   able to soar for 1060 h. For gliding speeds ranging
   from approximately 9 to 13 km/hr, the theoretical
   maximum range, without pauses for feeding, would
   fall between 9500 and 13 800 km.”
   
   Gibo, David L., and Pallett, Megan J., “Soaring Flight of Monarch Butterflies, Danaus Plexippus (Lepidoptera: Danaidae), during the Late Summer Migration in Southern Ontario.” Canadian Journal of Zoology, July 1979. https://doi.org/10.1139/z79-180.
   
   9-13km/hr = 2.5-3.6 m/s
6. 2.5/.0056 = 446
   
   3.6/.0014 = 2571
   
   These numbers sound surprisingly high, but they are at least in line with Gibo & Pallett’s estimate for maximal distance possibly flown with the 140mg of fat in a butterfly, if it could soar for the entire time:
   
   “Consequently, if soaring requires approximately
   the basal level of metabolic expenditure, an average D. piesippus with an initial fat supply of 140 mg,
   which could fly under power for only 11 h, may be
   able to soar for 1060 h. For gliding speeds ranging
   from approximately 9 to 13 km/hr, the theoretical
   maximum range, without pauses for feeding, would
   fall between 9500 and 13 800 km.”
   
   Gibo, David L., and Pallett, Megan J., “Soaring Flight of Monarch Butterflies, Danaus Plexippus (Lepidoptera: Danaidae), during the Late Summer Migration in Southern Ontario.” Canadian Journal of Zoology, July 1979. https://doi.org/10.1139/z79-180.
   
   Our understanding is that this much soaring without the need for powered flight is implausible, but this efficiency figure should still be approximately correct for shorter distances where soaring is feasible.
7. Gibo and Pallett estimate cruising flight to be about one fourth as costly as sustained flapping flight:
   
   “Since profile drag is proportional to the square of the speed, doubling the speed increases the profile drag by four (Irving 1977). Consequently, it seems reasonable to assume that the slower, less energetic cruising flight of D. plexippus requires approximately 25% of the energy expenditure of vigorous flight.”
   
   Gibo, David L., and Pallett, Megan J., “Soaring Flight of Monarch Butterflies, Danaus Plexippus (Lepidoptera: Danaidae), during the Late Summer Migration in Southern Ontario.” Canadian Journal of Zoology, July 1979. https://doi.org/10.1139/z79-180.
   
   Given an estimate of 837 J/g⋅hr for sustained flapping flight (see below), this gives us 209 J/g⋅hr for cruising.
8. 209 J/g⋅hr * 0.6g = 125J/hr = 0.035 J/s
9. “However, during migration D.
   plexippus adults usually adopt a slower cruising
   flight at an airspeed of 18 km/hr (Urquhart 1960), a
   flying strategy that should result in less fuel being
   consumed as a result of reduced drag at lower
   airspeeds.”
   
   Gibo, David L., and Pallett, Megan J., “Soaring Flight of Monarch Butterflies, Danaus Plexippus (Lepidoptera: Danaidae), during the Late Summer Migration in Southern Ontario.” Canadian Journal of Zoology, July 1979. https://doi.org/10.1139/z79-180.
10. >5/0.035 = 143
11. We were not able to find the flight metabolic rate of monarch butterflies. Instead we use the flight metabolic rate of 200 calories per gram per hour (837 J/g⋅hr) that was measured as a minimum for another butterfly species during energetic flight, as Gibo and Pallet (1979) seem to think is somewhat reasonable, in their paper on soaring flight of monarch butterflies:
    
    “If we assume that the metabolic rate of D. plexippus during sustained flapping flight is only 200 cal/g h^-1, the minimal value determined by Zebe (1954) for another butterfly (Vanessa sp.)…”
    
    Gibo, David L., and Pallett, Megan J., “Soaring Flight of Monarch Butterflies, Danaus Plexippus (Lepidoptera: Danaidae), during the Late Summer Migration in Southern Ontario.” Canadian Journal of Zoology, July 1979. https://doi.org/10.1139/z79-180.
12. 837 J/g⋅hr * 0.6g = 502 J/hr = 0.14 J/s
13. Maximal observed in one study:
    
    “On some days, the butterflies were achieving estimated ground speeds of more than 50 km/hr.”
    
    Gibo, David L., and Pallett, Megan J., “Soaring Flight of Monarch Butterflies, Danaus Plexippus (Lepidoptera: Danaidae), during the Late Summer Migration in Southern Ontario.” Canadian Journal of Zoology, July 1979. https://doi.org/10.1139/z79-180.
    
    Also, airspeed during sustained flapping is around 40km/hr, and it seems unlikely that the motion of air can’t add a further 10km/hr to ground speed:
    
    “In D. plexippus vigorous flight of this type produces an airspeed
    of approximately 40 kmlh (Urquhart 1960).”
    
    Gibo, David L., and Pallett, Megan J., “Soaring Flight of Monarch Butterflies, Danaus Plexippus (Lepidoptera: Danaidae), during the Late Summer Migration in Southern Ontario.” Canadian Journal of Zoology, July 1979. https://doi.org/10.1139/z79-180.
    
14. >13.9/0.14 = 99
15. “The farthest ranging monarch butterfly recorded traveled 265 miles in one day.” [265 miles = 426km]
    
    “Monarch Butterfly Migration and Overwintering.” Accessed November 25, 2020. https://www.fs.fed.us/wildflowers/pollinators/Monarch_Butterfly/migration/index.shtml.
16. For instance, if it pessimistically takes 100 meters of powered flight to take off, then it would be .02% of the distance, so if it was 25x as much energy as usual for that distance, it would add .5% to the total energy use, which is far within the margin of error for this very rough calculation.
17. For instance, in this video of soaring butterflies, most of them appear to flap their wings occasionally: https://www.youtube.com/watch?v=a-8SPgG–6I

    Usa Monarch. Soaring MONARCH BUTTERFLIES at Their Mexico Migration Site, 2019. https://www.youtube.com/watch?v=a-8SPgG–6I.

18. “A description of the thermal soaring activity observed on September 7, a particularly favourable day, provides a clear picture of the relative importance of this type of flight. On this day we recorded the greatest proportion of soaring flights for any single day of observation.

    Soaring accounted for 1964 s or 83.5% of this time, and powered flight for 359 s or 15.3%.”

    Gibo, David L., and Pallett, Megan J., “Soaring Flight of Monarch Butterflies, Danaus Plexippus (Lepidoptera: Danaidae), during the Late Summer Migration in Southern Ontario.” Canadian Journal of Zoology, July 1979. https://doi.org/10.1139/z79-180.

19. “FoodData Central.” Accessed December 10, 2020. https://fdc.nal.usda.gov/fdc-app.html#/food-details/167755/nutrients.
20. “Each year in late summer and fall in southern Ontario, monarch butterflies, Danaus plexippus plexippus L., engage in migratory flight to the southern U.S.A. and Mexico…

    …Late summer monarchs average approximately 600mg.”
    
    Gibo, David L., and Pallett, Megan J., “Soaring Flight of Monarch Butterflies, Danaus Plexippus (Lepidoptera: Danaidae), during the Late Summer Migration in Southern Ontario.” Canadian Journal of Zoology, July 1979. https://doi.org/10.1139/z79-180.