Fruit Mass of Carica papaya L. from Cultivars Aliança and THB from the Width and Length of the Fruit

Papaya (Carica papaya L.) is a typical plant with a tropical climate, but also grown in subtropical regions. Using mathematical models well-adjusted al-lows with good precision to estimate characteristics of interest. The objective was to adjust an equation that estimates the fruit mass for each cultivar of papaya, Aliança and THB, using only one measure, length or width. The experiment was conducted in the municipality of Linhares in the state of Espírito Santo, Brazil. Seedlings were planted on the same day, spaced 3.6 × 1.5 m and in rows side by side. Initially, the equations were modeled, they were linearized and then the covariance analysis was performed in order to verify the possibility of an equation that would serve both cultivars. As the covariance was significant, it was necessary to develop equations for each cultivar. To obtain the growth equations, 350 fruits of cultivar Aliança and 550 of THB were used. The validation was performed with 50 fruits of each. The characteristics evaluated were the largest width (W in mm), the longest fruit length (L, in mm) and the observed mass (OM in g). The equations that best fit were those of the power model that use width (W) as an independent variable.


Introduction
Papaya (Carica papaya L.) belongs to the Caricaceae family and is a large and perennial herbaceous plant [1]. Typical of tropical climate, but also cultivated in the Mediterranean region, like Spain [2], for example. According to [3], papaya fruits are excellent sources of calcium, pro-vitamin A and vitamin C (ascorbic acid). [4] and [5] affirm that the concentration of vitamin C varies between cultivars and fruit ripening stages.
In 2018, an average of 38.91 tons of papaya per hectare were harvested in Brazil, in an area of 27,250 hectares, totaling 1,060,392 tons [6]. In 2019, Brazil exported 44,238.067 tons of papaya, with the three largest buyers being Portugal, Spain and Germany [7]. The production of this fruit in the country has great economic, social and nutritional importance. Crops are constantly renewed, with a cultivation cycle of about 18 to 24 months [8]. Therefore, it requires and employs a large amount of labor, generating many direct and indirect jobs.
Currently in Brazil, there are 58 papaya cultivars registered with the Ministério da Agricultura, Pecuária e Abastecimento [9]. The cultivars Aliança and THB are both registered, part of the "Solo" group. In this group, "Solo", most cultivars belong, which have pyriform fruits, of small size and weight ranging from approximately 300 to 650 g [10].
The fruit growth curve can point to the most critical phase in its development and thus highlight the period of greatest demand for nutrients [11] and still allow an estimate of production [12].
In their study using the equatorial diameter of the blueberry fruit (Vaccinium spp.), using a power-type equation, [13] estimated its mass, [14] estimated an equation that, based on the diameters of the babassu fruit, can estimate its volume. [15] adjusted a quadratic equation that estimates the mass of pear (Pyrus communis L.) using the diameter of the fruit.
There are also some studies of papaya fruit growth. [16] related the diameter, the length of papaya fruits and the growth rate with degrees accumulated days.
[17] related the papaya growth to the number of days required until harvest.
And yet [18] they adjusted the quadratic equation for the growth of mountain papaya (Vasconcellea cundinamarcensis B.), which is a species that also belongs to the family Caricaceae.
No studies were found in the literature on papaya fruit growth in which they related the length and/or width of the fruit to its mass. The objective of this work was to adjust an equation that estimates the fruit mass for each papaya cultivar Aliança and THB, which uses only one measure, length or width.

Material and Methods
The experiment was carried out at Fazenda Santa Terezinha of the company Caliman Agrícola SA, located in the municipality of Linhares, Northern Espírito Santo State, Brazil, with latitude 19˚11'49''S, longitude 40˚04'20''W and altitude of 33 m. The region's climate is of the tropical Aw type by the Köppen classification, with rain in the summer and drought in the winter [19].
For the preparation of the experimental area, plowing, harrowing and the ne- The mathematical models tested for obtaining the growth curve were linear first degree (Equation (1)), potential (Equation (2)) and exponential (Equation which is a necessary step in the analysis of covariance. where OM is the observed mass (dependent variable) as a function of X (independent variable) which is the largest width (W) and the length (L). Six equations were obtained, two from each model for each cultivar, using the least squares method, and their respective coefficients of determination (R 2 ).  (4)), the root mean square error (RMSE) (Equation (5)) and the Willmott d index [20] (Equation (6)) were also determined.
The criteria for defining the best equation were: linear coefficient ( 0 β ) not different from zero; slope ( 1 β ) not different from one; MAE and RMSE closer to zero; determination coefficient (R 2 ) and Willmott d index [20] closest to one.
Descriptive statistics of the fruits were also performed. The making of graphs and statistical analyzes were performed using the R software [21] using the Expdes.pt package version 1.2 [22].

Results and Discussion
In Table 1, it can be seen that papaya fruits of various sizes were used to model the equations. The coefficient of variation (CV), which is a measure of variability, is considered very high for all measures (width, mass and length) in both cultivars according to the classification by [23] and [24] respectively. In studies of growth modeling, it is necessary to use elements of all sizes, so that it represents growth in the most realistic way possible, thus obtaining more accurate equations. This can be seen by a very high coefficient of variation.
The adjusted equations in the modeling are shown in Table 2. It can be seen that for the same type of equation, those that use the width of the fruit (W) as an independent variable have a higher coefficient of determination (R 2 ) compared to those that use the length (L). Given this fact, it can be seen that the width of the papaya fruit has a greater relationship with the mass. [15] in their study with pear state that the best adjustment in relation to the width may be associated with the piriform characteristic of the fruits, where there is a greater accumulation of mass in the medial-basal region, favoring the relationship between these characteristics. In other growth studies, the equations that best fit with the mass were also those that used the width of the fruit [13] [15]. The analysis of covariance was significant for the linear and angular coefficients (p < 0.05), showing that the two cultivars show different growth and, therefore, there was a need to adjust equations for each cultivar separately.
The equation must be chosen not only by the high coefficient of determination (R 2 ), but by several criteria as well as performed by [25] and [15]. The criteria for the validation of the equations are presented in Table 3. For the two cultivars, the model that met all adjustment norms was the potential that uses fruit  An equation that can accurately estimate the mass of a fruit using only one measure, in this case, the width, and yet without detaching it from the parent plant, is a good tool in growth and physiological monitoring studies, as evaluations require methods non-destructive. The producer can also use these equations to estimate the mass of fruits and still obtain an approximation of production. The use of potential equations is much simpler and faster to solve compared to non-linear models, especially if the researcher or the producer is in the field with only the smartphone in hand.
It is possible to estimate the growth of the fruit using these equations. They still have an advantage in relation to the use of equations that need to use degree days, since they use only a linear measure of the fruit. The equations that require degrees days require planning to be used, since one must know how many degrees have been accumulated since the opening of the flowers or withering of the petals until the moment, so that only this can estimate the growth of the fruit.
In Figure 1    in millimeters. It is worth mentioning that when using regression models for estimates, the values should not extrapolate from those used in the construction of the regression equation [26].
In Figure 2, the relationship between the observed mass (OM) and the estimated mass (EM) was very close, with a very high determination coefficient (R 2 ) in both cultivars.

Conclusions
It was possible to estimate the mass of the fruits of Carica papaya L. from cultivar Aliança and THB.
To estimate the mass of the fruit of Carica papaya L. from the Aliança cultivar, the equation