Back in 2008, Panofsky gave an empirical formula, $T=11.5W2/3$, for turkey baking time, T, in hours vs turkey weight, W, in pounds, the so-called Panofsky formula or Panofsky constant. Compared to the previously existed recipes that are based on the simple linear relationship between turkey weight and baking time, the Panofsky formula provides a more accurate estimate for the baking time. For instance, a general guideline of 13–20 min/lb was widely recommended in all previous turkey baking recipes. In this work, we conduct a comprehensive study of the turkey baking process that leads to a mathematical derivation of the Panofsky formula under some approximations. We also generalize the Panofsky formula to define a general Panofsky formula, $T=1PW2/3$, where P is defined as the Panofsky constant. Under spherical approximations, we then apply an accurate physical solution of the heat transfer equation and use the rigorous solution with numerical methods to study the generalized Panofsky formula and the Panofsky constant. We found that the generalized Panofsky formula can be perfectly applied to all turkey baking scenarios for baking time calculations. Furthermore, we did a careful analysis of the Panofsky constant, which equals 1.5 in the original Panofsky formula. The dependency of the new Panofsky constant on thermal properties of the turkeys and other initial parameters of baking, e.g., initial and final center temperature of the turkeys, oven temperature, thermal conductivity, specific heat, and turkey’s density, was carefully analyzed and mapped out. The Panofsky constant, P, could vary from 1.1 to 1.9 depending on these thermal parameters.

## I. INTRODUCTION

To most physicists, it is an essential belief that all natural phenomena can be understood and interpreted in both qualitative and quantitative ways according to general physics principles.^{1–8} A great example the authors have been looking carefully into is the culinary arts.

Human beings have been cooking with a long history.^{1–4} Cooking provides such a rich set of opportunities and examples to learn and study science, in particular, physics.^{5,6} It is an ideal and easy-to-access tool for learning and studying physics, including performing both theoretical and experimental research studies. It is easy to perform even in a regular home setting. Understanding the science in cooking, supported with interesting home-kitchen laboratory experiments, generates enthusiasm and provides strong motivation for anyone who is interested in physics and science. Many factors could affect the result of making a successful dish. Those factors include but are not limited to cooking time, cooking temperature, heating method, and size of the food. Each recipe is an experiment; deviating from tradition may result in culinary disaster. An intrepid cook is fundamentally a scientist who needs to explore a wide range of variables through careful experimentation by systematically varying one parameter of the recipe at a time and observing the outcome. Understanding of the cooking process of food is crucial in understanding the underlying physical and chemical phenomena, which is key to optimizing the culinary quality in terms of process, texture, and flavor optimization, as well as safety of meat and other types of foods’ consumption.^{7,9}

In the past, many research studies have been performed, both experimentally and theoretically, and many publications’ references have highlighted the importance of physical modeling.^{8–21}

Varlamov *et al.*^{15,16} utilized a simplified heat transfer model to study pizza baking,^{15} boiling, steaming, and rinsing Chinese foods^{16} and derived the quadratic power relationship with the food dimension. Olszewski^{8} used a rigorous heat transfer equation to solve the short cylinder to study the baking time of cake and its dependence on the dimensions of the cake. Nelson *et al.*^{11} studied steak cooking with a two-dimensional mathematical model. Unsworth and Durate,^{17} Buay *et al.*,^{18} Roura *et al.*,^{19} and Williams^{20} all have studied boiled eggs based on solving the thermal diffusion equations. Chang *et al.* modeled heat transfer during roasting of unstuffed turkeys.^{21}

A great example of applying physical thinking to culinary arts is the story about how SLAC former director, Professor WKH “Pief” Panofsky, developed the formula for baking of turkeys.^{22}

People have been baking turkeys for a long time.^{22–32} In all previously existed recipes, from generations to generations, a “15–30 min/lb” recipe was widely adopted.^{22–32} It says that the baking time of a turkey is linearly proportional to the weight of the turkey. There are many variations of this linear recipe, which we can easily find in every turkey cook book or via online searching. A revised recipe may call that the time per pound rate varies according to the range of the turkey weight. As the turkey becomes larger and much heavier, for instance, over 20 lb, then the minutes per pound rate should go down slightly.^{22–32}

In 2008,^{22} according to the original note from Nicholas Panofsky, “There was a point in time when my grandfather [SLAC Director Emeritus] WKH “Pief” Panofsky was not satisfied with the cooking times for turkeys of “30 min/lb.” This is of course reasonable because the time for which a turkey should be cooked is not a linear equation. “So Pief Panofsky derived an equation based on the ratio between the surface area and the mass of a turkey. He determined that the cooking time for a stuffed turkey in a 325 °F oven is $T=11.5W2/3$, where T is the cooking time in hours and W is the weight of the turkey in pounds. The constant 1.5 was determined empirically.”^{22}

The so-called Panofsky formula, $T=11.5W2/3$, provides a more accurate estimate for the baking time compared to the previously existed recipes with the simple linear relationship. The power of 2/3 dependency on the turkey weight is relatively easy to understand. The power of 1/3 of the weight essentially converts the weight into a characteristic dimension of the turkey, such as the radius or diameter or length of the turkey, and we also know that, in principle, the cooking time has a power of 2 dependent on the linear dimension of the food piece, that is, doubling the size of the food would get a cooking time increased by four times.

However, the exact physics beyond the above brief description could not be easily found or available online or in the textbooks or scientific literature. The questions we have been asking are the following: (1) How sound is the Panofsky formula from a full and accurate physical consideration? (2) What kinds of assumptions and approximations have been applied in order to get to use the formula? (3) How could we understand the physics behind the constant 1.5? (4) How does this constant depend on the physical and thermal parameters of the turkey cooking process?

The motivation of this research is to use accurate physical modeling along with some approximations to deal with the cooking of turkeys in order to make the complicated real-life problem truly understandable. We try to simplify the physics but meanwhile do not lose the essentials, as evident from how close the calculated cooking time is related to the real-life data. We try to find a unified framework to treat the cooking in order to provide us a clear overall picture on how to calculate and address the cooking time and recipe, which depends primarily on the size of the food, heating profile (i.e., temperature), and food thermal properties.^{8–21}

In this paper, we conduct a comprehensive study of the turkey baking process that leads to a mathematical derivation of the Panofsky formula with some approximations. We also generalize the Panofsky formula to define a general Panofsky formula, $T=1PW2/3$, where P is defined as the Panofsky constant. Adopting the spherical approximations, we derive an accurate physical solution of the heat transfer equation. We then use the rigorous solution with numerical methods to study the generalized Panofsky formula and the Panofsky constant. We found that the generalized Panofsky formula can be perfectly applied to all turkey baking scenarios for baking time descriptions. Furthermore, we did a careful analysis of the Panofsky constant, which equals 1.5 in the standard Panofsky formula. The dependency of the new Panofsky constant on thermal properties of the turkeys and other initial parameters of baking, e.g., initial and final center temperature of the turkeys, oven temperature, thermal conductivity, specific heat, and turkey’s density, was carefully analyzed and mapped out. The Panofsky constant, P, could vary from 1.1 to 1.9 depending on these thermal parameters.

The outline of the article is as follows: First, we apply the spherical approximation and then derive the heat transfer partial differential equation and finally find the solution of the heat transfer equations. Then, we mathematically derive the Panofsky formula under some approximations. We then consider all the other shapes and irregularities with a defined shape factor. Next, after a careful survey of the thermal properties and parameters of turkeys, we carefully study the generalized Panofsky formula’s soundness and the dependency of the Panofsky constant on all the key physical and thermal parameters. In Sec. V, we touch upon the issues of weight loss and so forth.

## II. HEAT TRANSFER EQUATIONS

### A. General discussion

According to Fourier’s law,^{33–38} the heat flux q (in W/m^{2}), as shown schematically in Fig. 1, is proportional to the temperature gradient, i.e., q = −k$\u22c5dTdx$ for one-dimensional systems. For the three-dimensional system, $q\u20d7$ = −k · $\u2207$T, where $q\u20d7$ is a vector, $\u2207$ is the gradient, and k is the thermal conductivity in W/(cm K),

This leads to the one-dimensional heat diffusion equation

where T is a function of location and time, i.e., T(x, t), and $\alpha $ is the thermal diffusivity with the unit of m^{2}/s. $\alpha $ can be described as k/$\rho $c, with $\rho $ representing the density (kg/m^{3}) and c representing the specific heat [J/(kg K)].

In three-dimensions, the heat transfer equation becomes

where

### B. A sphere with azimuthal symmetry

For a sphere with azimuthal symmetry, as shown schematically in Fig. 2, during the heat transfer,^{2–18,18–25} we have $\u2202T\u2202\theta =0$ and $\u22022T\u2202\phi 2=0$, and the heat transfer equation becomes

If we apply *V* = *r* · *T* to the above equation, we get

We can decouple V(r, t) into

We get

and

Then, we have

It can be rearranged into

Since the left side is only r-dependent and the right side is only t-dependent and they are equal to each other, they must be neither r- or t-dependent. So, we have

Then, we have

and

From the above equation, we have

For

Now, we have

For cooking (heating) food starting with low temperature T_{0} and surrounded at a high temperature (bath temperature) T_{h}, we have the following boundary conditions:

We have

We then have

We define

Thus, we have

With r = 0, we can then get the temperature at the center of the sphere,

We can spell out the equation with some of the initial (and deciding) terms,

where $\tau =R2\pi 2\u22c5\alpha $ and $\alpha =k\rho c$.

### C. The Panofsky formula and the Panofsky constant

When we carefully look at Eq. (1), the value of the second term and also all the later terms in the parentheses goes down very quickly compared to the first term. The other higher order terms have even smaller contribution that can be ignored without causing any meaningful error in the calculations. In all our numerical calculations discussed later (i.e., Figs. 3–8), in order to build our confidence, we intentionally compared the calculation results with only the first term, the initial two terms, the initial three terms, the initial four terms, and the initial ten terms.

Based on our numerical calculation, in the case that we are confident that the first term dominates the result, we can ignore all the higher order terms other than the first term in the parentheses, and we get

We can re-arrange the equation into

since $\tau =R2\pi 2\u22c5\alpha $ or $\tau =L2\pi 2\u22c5\alpha $ and $\alpha =k\rho c$, we have

Thus, the new approximated equation with only the first term considered gives a square dependence on the critical dimension, i.e., the radius R for the sphere.

Now, let us convert Eq. (2) into a baking time dependency on the weight of the turkey, which is essentially the Panofsky formula or an approximation.

Since the weight of the turkey is *W* = *ρ* · *V*, where *V* is the volume $V=43\pi R3$, we have $W=43\pi \rho R3$. Thus, we have

where

We define the Panofsky constant, P, as

In the Panofsky constant formula (3), both *ρ* and *α* are in SI units.

With Eq. (3) and the standard^{39,40} density around 1050 kg/m^{3}, the specific heat c around 3530 J/(kg K), the thermal conductivity K around 0.5 W/(m K),^{39,40} the oven temperature at 163 °C (325 °F), the initial turkey temperature at 20 °C, and the center temperature of the spherical turkey at 85 °C (185 °F), our calculated P is 1.16, which is much smaller than 1.5, which Dr. Panofsky suggested in 2008 through his famous formula.

We do realize in the process of deriving Eq. (3) that all the terms except the first term in Eq. (1) are neglected. In Sec. IV, we will try to calculate the Panofsky constant and demonstrate the feasibility of the Panofsky formula through a rigorous numerical calculation with higher terms considered.

### D. Approximations for non-spherical shapes

Since the turkey is never a spherical shape, this section introduces a shape factor^{18} to discuss the possible deviation caused by our approximation with the shape.

We will use the other geometry shapes such as cube and short cylinder to provide some insights.

Because the received heat is proportional to the surface area and the received energy per volume is inversely proportional to the total volume, the inverse of the baking time (or heating time) is proportional to the surface area and inversely proportional to the volume of the piece. We then have

For a sphere, we have

For a cube with a side length of a, we have

If the cube has the same volume as the sphere, $43\pi R3=a3$, we have $12a\u22450.8R$. That is, from thermal transfer (heating) perspective, a cube with the same volume as a sphere is equivalent to a sphere with 0.8 (80%) of the diameter, i.e., *a* ≅ 0.8 · (2*R*).

For a short cylinder (such as a cake) with a radius r and height h,

With the volume of the short cylinder the same as the sphere, $43\pi R3=\pi r2h=\pi r3forh\u223cr,wegetr\u22451.1R$, so we have 0.75*r* = 0.75 · 1.1*R* ≈ 0.83*R*.

That is, from the thermal transfer (heating) perspective, a short cylinder (radius and height similar) with the same volume as a sphere is equivalent to a sphere with 0.83 (83%) of the radius.

From another perspective, the above analysis also proves that the sphere is the most efficient geometry in heat transfer. In other words, with the same weight and density, the shape deviation from a perfect sphere would result in a slower rise (longer heating time) in the center temperature, i.e., a less efficient heating process. We will come back to this discussion further in Sec. IV A when we discuss the generalized Panofsky formula.

## III. TURKEY’S THERMAL PARAMETERS

In our work, a through literature survey of the thermal properties of turkeys was conducted. However, their thermal parameters vary hugely in some of the reported research studies, as listed in Table I.

Parameter . | Unit . | Value . | Condition or trend . | References . |
---|---|---|---|---|

Density | kg/m^{3} | 1050 | 26 and 27 | |

1070 | 20 °C | 46 | ||

∼1000 | 80 °C | 46 | ||

Thermal conductivity | W/(m K) | 0.5 | 39 and 40 | |

0.464 | 47 | |||

0.3 | 20 °C | 46 | ||

0.5 | 60–80 °C | 46 | ||

Specific heat | J/(kg K) | 3530 | 39 and 40 | |

2850 | 30 °C | 46 | ||

3380 | 80 °C | 46 | ||

3450 | 20 °C | 48 |

According to Refs. 39–51, which summarize all the essential thermal parameters for all the foods, turkey has a reported density around 1050 kg/m^{3}, the specific heat c around 3530 J/(kg K), and the thermal conductivity K around 0.5 W/(m K).^{39,40}

According to Ref. 46, with temperature between 20 and 60 °C, the thermal conductivity K increases linearly from 0.3 to 0.5 W/(m K). Between 60 and 80 °C, it remained constant. From 20 to 40 °C, the density of the turkey decreases slightly as a function of temperature. A transition phase was observed from 40 to 60 °C, which was followed by a decrease from 60 to 80 °C. There was a decrease of about 50 kg/m^{3} between the density of a raw product at room temperature (at maximum 1070 kg/m^{3}) and the product heated to 80 °C (at minimum 970 kg/m^{3}) due to the gelation or setting of the structure.

For the specific heat, the heat capacity increased linearly from 2850 to 3380 J/(kg K) for temperature from 30 to 80 °C. The heat capacity includes the enthalpy change as a result of the chemical reactions from the cooking process.

In Ref. 47, the thermal conductivity K is 0.464 W/(m K) for turkeys.

In Ref. 48, the specific heat c is 3450 J/(kg K) at 20 °C.

The temperature dependence of specify heat and thermal conductivity is partially due to the change of composition in the turkey in the heating process.

## IV. NUMERICAL CALCULATIONS

### A. The baseline case

In this section, we have done the numerical calculations based on the rigorous equation (1). As the baseline, we adopt the thermal parameters from Refs. 39 and 40, that is, turkey has a density around 1050 kg/m^{3}, the specific heat c around 3530 J/(kg K), and the thermal conductivity K around 0.5 W/(m K).^{39,40} We also use the baseline with the oven temperature set at constant 163 °C (325 °F) and initial turkey temperature at 20 °C.

A typical turkey weighs between 8 lb (3.63 kg) and 25 lb (11.3 kg). We approximate the turkey with a spherical shape. Based on the density and weight, we can calculate the effective radius of the turkey based on the weight. We have radius at 115 mm for 6.8 kg, 127 mm for 9.1 kg (20 lb), and 137 mm for 11.3 kg.

We calculate the cooking time based on reaching a center temperature of 85 °C (185 °F). Table II shows the calculated results.

Weight (kg) . | Effective radius (mm) . | Time taken to reach 85 °C at the center . |
---|---|---|

3.63 (8 lb) | 93.8 | 141 min (2.35 h) |

4.54 (10 lb) | 101 | 163 min (2.72 h) |

6.8 (15 lb) | 115 | 212 min (3.53 h) |

9.07 (20 lb) | 127 | 258 min (4.31 h) |

11.33 (25 lb) | 137 | 302 min (5.03 h) |

Weight (kg) . | Effective radius (mm) . | Time taken to reach 85 °C at the center . |
---|---|---|

3.63 (8 lb) | 93.8 | 141 min (2.35 h) |

4.54 (10 lb) | 101 | 163 min (2.72 h) |

6.8 (15 lb) | 115 | 212 min (3.53 h) |

9.07 (20 lb) | 127 | 258 min (4.31 h) |

11.33 (25 lb) | 137 | 302 min (5.03 h) |

We plot out the calculated time (in minutes) with the turkey’s weight (in pounds). It is displayed in blue dots in Fig. 3.

In Fig. 3, the dots are the calculated data. The dotted line is a power function fitting to the calculated data. It shows that the cooking time has a 0.667 (2/3) power relationship with the weight of the turkey, with a fitting equation $T(hours)=0.5862\u22c5W2/3=11.71W2/3$ (W is the weight in pounds), with the fitting *R*^{2} = 1.0.

Our fitting formula of the turkey baking time (in hours) in relation to the weight of the turkey (in pounds)

is very close to the Panofsky formula, which is $T(hours)=11.5W2/3$.^{22}

The constant of 1/1.5 in the Panofsky equation or formula was determined empirically.^{22}

However, our fitting formula is fitted to all theoretical calculations based on the approximation to treat turkeys as a spherical body from a simple heat transfer perspective. It is amazing that we could fit all the strictly calculated data so neatly with this simple formula but generalized the so-called Panofsky constant.

The above calculated times match well with the normal recommended turkey baking time (Table III) from, for example, http://www.csgnetwork.com/turkeycookingtimecalc.html. Our calculated times also match well with our own turkey baking times conducted in our own experiments.

Turkey weight (pound) . | Cooking time in a 325 °F oven (hour) . | Internal temperature (°F) . |
---|---|---|

6–8 | 2.25–2.75 | 185 |

8–12 | 2.75–3.25 | 185 |

12–16 | 3.25–4.25 | 185 |

16–20 | 4.25–4.75 | 185 |

20–24 | 4.75–5.25 | 185 |

Turkey weight (pound) . | Cooking time in a 325 °F oven (hour) . | Internal temperature (°F) . |
---|---|---|

6–8 | 2.25–2.75 | 185 |

8–12 | 2.75–3.25 | 185 |

12–16 | 3.25–4.25 | 185 |

16–20 | 4.25–4.75 | 185 |

20–24 | 4.75–5.25 | 185 |

Our rigorous calculation based on the baseline thermal parameters gives a Panofsky constant at 1.71, which is closer but larger than the 1.5.

In the following study, we generalize the Panofsky formula into

and we define P as the so-called Panofsky constant. For the standard Panofsky formula, the Panofsky constant is 1.5. For our above baseline case, P has a value of 1.71. In the following, we conduct a careful study of the Panofsky constant and its dependency on other parameters.

In Sec. II D, we indicate that the sphere is the most efficient geometry in heat transfer. With the same weight and density, the shape deviation from a perfect sphere would result in a slower rise (longer heating time) in the center temperature, i.e., a less efficient heating process. It means that the Panofsky constant, P, will be an upper limit when we approximately consider the turkey with the same weight as a perfect spherical shape. After considering the real shape, which is deviated from the sphere, we should re-write the Panofsky formula into

where *P*_{real} is the Panofsky constant for real turkeys, *P*_{s} is the Panofsky constant with an ideal spherical shape, and ∈ is the shape correction factor. Since the sphere is the most efficient shape in heating, ∈ is always equal to or less than 1.0.

### B. The Panofsky constant P dependency on other parameters

#### 1. The initial temperature of the turkey

The initial temperature of the turkey at 20 °C is our baseline. Obviously, if this temperature is different, then the calculated Panofsky parameter would change. The gray dots in Fig. 3 are for an initial temperature of 10 °C. It gives a Panofsky constant P = 1.62. Figure 4 shows the dependency of the calculated Panofsky constant vs the initial temperature.

#### 2. The oven temperature

The orange dots in Fig. 3 are for the oven temperature at 177 °C (350 °F). The calculated Panofsky constant is P = 1.82 with all the other parameters as used in the baseline case.

#### 3. The impact of thermal conductivity

We find that the Panofsky constant is very sensitive to the value of the thermal conductivity. With the thermal conductivity varying from 0.4 to 0.5 W/(m K), the Panofsky constant increases from 1.36 to 1.71, as shown in Fig. 5. All the other parameters are the same as the baseline cases. From the fitting in Fig. 5, it shows that the Panofsky constant is in a linear relationship with the thermal conductivity, K. This result agrees well with Eq. (3), which is derived with an approximation from Eq. (1).

#### 4. The impact of specific heat

We find that the Panofsky constant also is very sensitive to the value of the specific heat. With the specific heat varying from 2800 to 3980 J/(kg K), the Panofsky constant decreases from 1.93 to 1.36, as shown in Fig. 6. In Fig. 6, all the other parameters are the same as in the baseline cases except that the thermal conductivity of 0.45 W/(m K) rather than 0.5 W/(m K) is used. From the fitting in Fig. 6, it shows that the Panofsky constant is in an inverse relationship with the specific heat c. This result also agrees well with Eq. (3) that is derived with an approximation from Eq. (1).

#### 5. The impact of density

The dependency of the Panofsky constant on the density is more complicated than what it seems to be. From Eq. (3), it seems that we should find a power of (−1/3) dependency between the Panofsky constant K and the density *ρ*. Our calculation results shown in Fig. 7 show that the Panofsky constant decreases from 1.56 to 1.51 with the increase in the density from 1000 to 1100 kg/m^{3}. However, the dependency is not in a power of (−1/3) relation based on the fitting in Fig. 7. The reason for that is that the change in the density also changes the weight if the dimension of the turkey does not change. Or if the weight is kept the same, then the dimension would change.

#### 6. The rate of cooking time per pound

All the turkey baking recipes prior to the Panofsky formula relied on a simple rate number, which is the cooking time (in minutes) per pound of the turkey. In the following, we look at this rate in more detail based on our numerical calculation.

Figure 8 shows a set of calculated results with the following parameters: 1050 kg/m^{3} for density, 3600 J/(kg K) for the specific heat, 0.45 W/(m K) for the thermal conductivity, 163 °C (325 °F) for the oven temperature, 20 °C for the initial turkey temperature, and 85 °C for ending center temperature. Figure 8(a) shows the calculated cooking time as a function of the weight, which is in a perfect Panofsky formula. The fitting gives a Panofsky constant of 1.5. Figure 8(b) is the calculated average cooking time per pound. It shows that the average cooking time per pound is at 25 min/lb for a 4-lb turkey and then goes down to around 15 min/lb for a 20-lb turkey and then further goes down to about 13 min/lb for a 30-lb turkey. Those values are in line with all the numbers recommended in the existing recipes. Figure 8(c) is the calculated instant rate of cooking time per pound, that is, the slope of the curve in Fig. 8(a). It says that for a turkey of 5 lb, its cooking time increases about 14.5 min for a 1-lb increase in weight. This rate is about 10 min/lb for a 20-lb turkey.

## V. DISCUSSIONS ABOUT COOKING LOSS AND OTHER FACTORS

Since all foods contain a significant amount of water and some fat, naturally, during a high temperature process, the cooking process leads to cooking loss in weight and food shrinkage as a result.

Depending on food types and cooking methods, the cooking process resulted weight loss or food shrinkage happen at different rates. An oven-baked turkey also shrinks and loses much less than an openly roasted Peking duck. For a turkey baked by a typical closed oven method, the weight loss is not significant.^{52} Thus, we did not consider the impact of weight loss and shrinkage issue.

## VI. CONCLUSION

Back in 2008, Panofsky gave an empirical formula, $T=11.5W2/3$, for turkey baking time T in hours vs turkey weight W in pounds, the so-called Panofsky formula or Panofsky constant. Compared to the previously existed recipes that are based on the simple linear relationship between turkey weight and baking time, the Panofsky formula provides a more accurate estimate for the baking time. For instance, a general guideline of 13–20 min/lb was widely recommended in all previous turkey baking recipes. In this work, we conduct a comprehensive study of the turkey baking process that leads to a mathematical derivation of the Panofsky formula under some approximations. We also generalize the Panofsky formula to define a general Panofsky formula, $T=1PW2/3$, where P is defined as the Panofsky constant. Under the spherical approximations, we then apply an accurate physical solution of the heat transfer equation and use the rigorous solution with numerical methods to study the generalized Panofsky formula and the Panofsky constant. We found that the generalized Panofsky formula can be perfectly applied to all turkey baking scenarios for baking time calculations. Furthermore, we did a careful analysis of the Panofsky constant, which equals 1.5 in the original Panofsky formula. The dependency of the new Panofsky constant on thermal properties of the turkeys and other initial parameters of baking, e.g., initial and final center temperature of the turkeys, oven temperature, thermal conductivity, specific heat, and turkey’s density, was carefully analyzed and mapped out. The Panofsky constant, P, could vary from 1.1 to 1.9 depending on these thermal parameters.

## ACKNOWLEDGMENTS

L. R. Wang and Y. J. Jin would like to acknowledge both The Pennington School and Phillips Academy, Andover, for their support throughout this study.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

https://www.allrecipes.com/article/turkey-cooking-time-guide/ [20 min/lb general rule, 350 °F (175 °C) oven, 10–18 lb for 3–3.5 h, 18–22 lb for 3.5–4 h, 22–24 lb for 4–4.5 h, and 24–29 lb for 4.5–5 h for unstuffed turkeys, and add 0.75–1.25 for stuffed turkeys. Interior temperature is 75 °C].

https://www.southernliving.com/thanksgiving/turkey-cooking-time-per-pound (325 °F for oven, 8–12 lb, 2.75–3 h, 12–14 lb, 3–3-3.75 h, 14–18 lb, 3.75–4.25 h, 18–20 lb, 4.25–4.5 h, and 20–24 lb, 4.5–5 h for unstuffed turkeys).

https://www.delish.com/holiday-recipes/thanksgiving/a44899/how-long-cook-turkey/ (325 °F oven, 8 lb, 2.75 h, 10 lb, 2.9 h, 12 lb, 3 h, 14 lb, 3.75 h, 16 lb, 4 h, 18 lb, 4.25 h, 20 lb, 4.5 h, 22 lb, 4.75 h, and 24 lb, 5 h for unstuffed turkeys. For stuffed one, add more time).

https://www.huffpost.com/entry/how-long-to-cook-a-turkey-per-pound_n_56566cc5e4b072e9d1c1c469 (USDA minimal 165 °F for interior, 325 °F oven, 8–12 lb, 2.75–3 h, 12–14 lb, 3–3.75 h, 14–18 lb, 3.75–4.25 h, 18–20 lb, 4.25–4.5 h, and 20–24 lb, 4.5–5 h for unstuffed turkeys).

https://www.epicurious.com/holidays-events/the-easiest-way-to-cook-turkey-article (12–14 lb, 350 °F for 2.75–3 h; 325 °F for 3–3.75 h; 15–16 lb, 350 °F for 3.5–3.75 h, 325 °F for 3.75–4 h; 18–20 lb, 350 °F for 4–4.5 h, 325 °F for 4.25–4.5 h; 21–22 lb, 350 °F for 4.5–4.75 h, 325 °F for 4.75–5 h; 24 lb, 350 °F for 4.75–5 h, and 325 °F for 5–5.25 h).

https://www.bhg.com/recipes/how-to/handling-meat/how-long-to-cook-a-turkey/ (325 °F. 8–12 lb, 2.75–3 h; 12–14 lb, 3–3.75 h; 14–18 lb, 3.75–4.25 h; 18–20 lb, 4.25–4.5 h; and 20–24 lb, 4.5–5 h).

https://dinersjournal.blogs.nytimes.com/2011/11/24/how-long-should-i-cook-my-turkey/ (325 °F oven, unstuffed, 15 min/lb. 350 °F oven, 13 min/lb).

https://parade.com/236361/smccook/how-long-to-cook-the-perfect-turkey-and-how-to-know-when-a-turkey-is-done/ (325 °F oven, unstuffed, 8–12 lb, 2.75–3 h; 12–14 lb, 3–3.75 h; 14–18 lb, 3.75–4.25 h; 18–20 lb, 4.25–4.5 h; and 20–24 lb, 4.5–5 h).

https://www.countryliving.com/food-drinks/a29762210/how-long-to-cook-turkey/ (325 °F, unstuffed, 15 min/lb).

Composition data from USDA (1996). Initial freezing point data from Table I in Chap. 30 of the 1993 ASHRAE Handbook—Fundamentals. Specific heats calculated from equations in this chapter. Latent heat of fusion obtained by multiplying water content expressed in decimal form by 334 kJ/kg, the heat of fusion of water (Table I in Chap. 30 of the 1993 ASHRAE Handbook—Fundamentals).