Cretoxyrhina the fastest shark ever?

[Macrophyseter]

The following content is purely based on applying a mathematical equation from Ferron (2017), who obtained it from Sambilay (1990). No other source besides a few used for length measurements have been used besides Ferron (2017). I am not a mathematician and my calculations may have mistakes, which is why I've shown all of my work in case any of you want to check my math. The formulas themselves may possibly be updated/invalidated by later studies. Please take this with the finest grain of salt.

 

Apparently, Cretoxyrhina may possibly be the one of the, if not the fastest shark known so far according to a set of equations. When reading through the scientific paper Ferron (2017), I found out that the equation he used to make his speed calculations are ones that I might be able to do myself, so here I am making a post that consists of a wall of algebraic nonsense to try to see for myself on things.

 

Ferron (2017) used an equation that was developed by Sambilay (1990) (which I unfortunately cannot access) to calculate the maximum burst speed of a large fish:

 

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ Log₁₀(PCL) + 0.3478 ⋅ Log₁₀(AR)

 

AR stands for aspect ratio, which is the ratio between the height and surface area of the caudal fin.

PCL stands for post-caudal length and can be calculated from a shark's total length (TL) using this equation:

 

PCL = -0.9195 + 0.8535 ⋅ TL

 

When Ferron (2017) inputted the equation for Cretoxyrhina mantelli using Shimada (2006)'s conservative estimates for maximum size of 640 cm as total length (TL) and an unspecified aspect ratio (AR), a burst swimming speed of ≈70 km/h was calculated. This is essentially one of the highest confirmed calculated speeds of any shark ever recorded. I decided to try the equation for myself, which is shown below. I assumed that the AR used would be the conservative aspect ratio of 4.3 (The aspect ratios for C. mantelli were 4.3 and/or 4.9 depending on the type of angle used).

 

AR = 4.3

PCL = -0.09195 + 0.8535 ⋅ TL

PCL = -0.09195 + 0.8535 ⋅ (640)

PCL = 546.14805

 

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ Log₁₀(PCL) + 0.3478 ⋅ Log₁₀(AR)

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ Log₁₀(546.14805) + 0.3478 ⋅ Log₁₀(4.3)

Log₁₀(546.14805) = x₁

x₁ = Log(546.14805)/Log(10)

x₁ = (2.73731038736)/(1)

x₁ = 2.73731038736

Log₁₀(4.3) = x₂

x₂ = Log(4.3)/Log(10)

x₂ = (0.6334684556)/(1)

x₂ = 0.6334684556

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ (2.73731038736) + 0.3478 ⋅ (0.6334684556)

Log₁₀(S_b) = 1.84995107

S_b = 10^1.84995107

S_b = 70.7866028

 

S_b ≈ 71 km ⋅ h^-1

S_b ≈ 44 mi ⋅ h^-1

 

My result is a burst swimming speed of 70.8 km/h, 71 km/h when using proper rounding. Sort of off, but considering that the aspect ratio could have been different, I think this is good enough. But Ferron (2017) only used Shimada (2006)'s conservative estimates. Here, I redid the equation but used the maximum size of 700 cm as total length (TL)  and a mean estimate of 4.6 for aspect ratio (AR)-

 

AR = 4.6

PCL = -0.09195 + 0.8535 ⋅ TL

PCL = -0.09195 + 0.8535 ⋅ (700)

PCL = 597.35805

 

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ Log₁₀(PCL) + 0.3478 ⋅ Log₁₀(AR)

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ Log₁₀(597.35805) + 0.3478 ⋅ Log₁₀(4.6)

Log₁₀(597.35805) = x₁

x₁ = Log(597.35805)/Log(10)

x₁ = (2.7762347206)/(1)

x₁ = 2.7762347206

Log₁₀(4.6) = x₂

x₂ = Log(4.6)/Log(10)

x₂ = (0.6627578317)/(1)

x₂ = 0.6627578317

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ (2.7762347206) + 0.3478 ⋅ (0.6627578317)

Log₁₀(S_b) = 1.8847622067

S_b = 10^1.8847622067

S_b = 76.6941443789

 

S_b ≈ 77 km ⋅ h^-1

S_b ≈ 48 mi ⋅ h^-1

 

My results show that a C. mantelli that measures 7 meters in length can swim at a bust speed of almost 77 km/h. That is essentially faster than any ocean animal except some of the billfishes according to popular estimates. Amazingly, this may possibly not be too farfetched. It has already been predicted for a long time that C. mantelli is likely a fast swimmer, potentially one of the fastest. Shimada (1997) showed that Cretoxyrhina possess a scale pattern that has superb efficiency in drag reduction (such patterns have only been found in the fastest Lamnids today). Kim et al. (2013) noted that the caudal fin of Cretoxyrhina represents the most extreme case of a Type 4 fin, where both the Cobb's and hypochordal ray angles are higher than any known shark. Type 4s are ones that are the most efficient in building speed, so maybe the "most Type 4" of them all could be the fastest?

 

 

 

For comparison, I also applied the equation to Carcharodon carcharias, which is most similar with C. mantelli in morphology and ecological role (I assumed that the largest confirmed length is 610 cm)

 

AR=2.7*

PCL = -0.09195 + 0.8535 ⋅ TL

PCL = -0.09195 + 0.8535 ⋅ (610)

PCL = 520.54305

 

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ Log₁₀(PCL) + 0.3478 ⋅ Log₁₀(AR)

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ Log₁₀(520.54305) + 0.3478 ⋅ Log₁₀(2.7)

Log₁₀(520.54305) = x₁

x₁ = Log(520.54305)/Log(10)

x₁ = (2.7164566524)/(1)

x₁ = 2.7164566524

Log₁₀(2.7) = x₂

x₂ = Log(2.7)/Log(10)

x₂ = (0.4313637642)/(1)

x₂ = 0.4313637642

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ (2.7164566524) + 0.3478 ⋅ (0.431367642)

Log₁₀(S_b) = 1.7672462077

S_b = 10^1.7672462077

S_b = 58.5121703926

 

S_b ≈ 59 km ⋅ h^-1

S_b ≈ 36 mi ⋅ h^-1

 

As far as I know, the fastest recorded speed for a great white is about 56 km/h. Considering that they probably have been clocked from smaller, more average individual, this calculation seems reasonable.

 

However, if I input the equation to Isurus oxyrinchus using the 2013 Huntington Beach catch record of 373 cm for total length (TL)-

 

AR=3.3*

PCL = -0.09195 + 0.8535 ⋅ TL

PCL = -0.09195 + 0.8535 ⋅ (373)

PCL = 318.26355

 

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ Log₁₀(PCL) + 0.3478 ⋅ Log₁₀(AR)

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ Log₁₀(318.26355) + 0.3478 ⋅ Log₁₀(3.3)

Log₁₀(318.26355) = x₁

x₁ = Log(318.26355)/Log(10)

x₁ = (2.502786903)/(1)

x₁ = 2.502786903

Log₁₀(3.3) = x₂

x₂ = Log(3.3)/Log(10)

x₂ = (0.51851399)/(1)

x₂ = 0.51851399

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ (318.26355) + 0.3478 ⋅ (0.51851399)

S_b = 10^2.086400122

S_b = 122.0113188

 

S_b ≈ 122 km ⋅ h^-1

S_b ≈ 76 mi ⋅ h^-1

 

-I get a calculated burst speed of 122 km/h, completely tearing the "Cretoxyrhina the fastest shark" claim to shreds. Mako sharks already are the fastest of all extant sharks, but having a calculated speed of this much is mind-blowing. As far as I know, the fastest recorded speed for a mako shark (and therefore any shark) is about 74 km/h, but my calculation is higher than that of any ocean animal ever recorded except the sailfish whose swimming speeds is up to 130 km/h according to some sources. However, it is possible that I either messed up the math, this equation might not work on smaller fishes, or this equation calculates the dire, dire fastest speed of a large fish. If the latter is true, I'm really eager to obtain an aspect ratio for the sailfish and run it through the equation and see if it tops the peregrine falcon. 

 

 

What do you guys think?

 

 

*AR was obtained from Ferron (2017)

EXTRAS

 

 

Equation for a shark's cruising speed

 

Log₁₀(S_c) = -0.828 + 0.619 ⋅ Log₁₀(PCL) +0.3478 ⋅ Log₁₀(AR)

 

 

Extra input

 

AR = 4.6

PCL = -0.09195 + 0.8535 ⋅ TL

PCL = -0.09195 + 0.8535 ⋅ (640)

PCL = 546.14805

 

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ Log₁₀(PCL) + 0.3478 ⋅ Log₁₀(AR)

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ Log₁₀(546.14805) + 0.3478 ⋅ Log₁₀(4.6)

Log₁₀(546.14805) = x₁

x₁ = Log(546.14805)/Log(10)

x₁ = (2.73731038736)/(1)

x₁ = 2.73731038736

Log₁₀(4.6) = x₂

x₂ = Log(4.6)/Log(10)

x₂ = (0.6627578317)/(1)

x₂ = 0.6627578317

Log₁₀(S_b) = -0.0659 + 0.6196 ⋅ (2.73731038736) + 0.3478 ⋅ (0.6627578317)

Log₁₀(S_b) = 1.86064469

S_b = 10^1.86064469

S_b = 72.55121501

 

S_b ≈ 72 km ⋅ h^-1

S_b ≈ 45 mi ⋅ h^-1
 

 

[MikaelS]

Cretoxyrhina mantelli could probably reach closer to 8 m in TL based on isolated teeth from the Turonian of UK. One of the incomplete syntypes from the Lewis area (an isolated crown of an anterior tooth) would have been about 80 mm high originally. This is about 25% larger than the largest anteriors from the Western Interior seaway.

[Macrophyseter]

EDIT:

 

I've realized that I've made a miscalculation in the Isurus oxyrinchus application (I wrote the actual PCL instead of log₁₀(PCL) for the latter's solution). While recalculating it, I ended up with an S_b of 46.255 km ⋅ h^-1.