Let us first quantify the power density that the sun projects onto the sensor.
For the purpose of this calculation, let's assume a focal length where the projection of the sun is exactly 1 mm² in area. With an angular solar diameter of 0.53°, this occurs at about 121 mm focal length. Now if we project the full sun with a power (solar constant S₀) of
1400 W/m², focused with an 121 mm f/1.8 lens (collecting diameter is 121 mm/1.8 = 67.2 mm, meaning a collecting area of 3550 mm² = 0.00355 m²), onto 1 mm² = 0.000001 m² of the sensor, we get a deposited amount of heat of almost exactly 5 MW/m²: 1400 W/m²⋅0.00355 m² / 0.000001 m² = 4970000 W/m² ≈ 5 MW/m². This power density is independent of focal length (this specific value is for f/1.8), and is proportional to 1/(f-number)². For a 1 mm² spot, this equates to a heating power of 5 W = 5 J/s.
Here is the solar power density on the sensor for different f-numbers:
| f/1.2 | f/1.8 | f/2.8 | f/4 | f/5.6 | f/8 | f/11 |
| 11 MW/m² | 5 MW/m² | 2 MW/m² | 1 MW/m² | 520 kW/m² | 250 kW/m² | 130 kW/m² |
[Table 1]
Let's next consider the sensor.
The detector itself is a CMOS die made of silicon. While silicon only melts at around 1700 K, the sensor is a
composite of several other materials in addition to the silicon detector area. The composite construction contains metal traces that melt at 900 K for aluminium and 1400 K for copper, adhesives (500-600 K for epoxy- and silicone-based adhesives) for bonding, and also organic polymer (plastics) for the Bayer filter with a melting point of just 500-600 K. As such, the photosites should already take damage at above ~600 K (330 °C), where the traces start to delaminate and photosites become unreadable (hot or dead pixels).
The sensor is around
1 mm thick. Silicon, which makes the majority of the sensor, has a specific heat capacity of around 0.70 J/(g⋅K) and a density of 2.3 g/cm³, so a volumetric heat capacity of 1.6 J/(cm³⋅K). That of the metal traces is a bit higher, that of the adhesives is a bit lower, so let's just assume an overall heat capacity of the sensor of 1.6 J/(cm³⋅K). Thus, a 1 mm² area of the sensor has a heat capacity of around 0.0016 J/(mm²⋅K), and the total sensor has a heat capacity of 0.53 J/K for an 22.3 mm⋅14.9 mm = 332.3 mm² APS-C sensor and 1.38 J/K for a 36 mm⋅24 mm = 864 mm² full-frame sensor.
Let us quickly consider the case of omitting any heat transfer (which is unphysical, but just for illustrative purposes), and calculate, the maximum time to heat the sun-illuminated area of the sensor by 300 K (from ambient ~300 K to 600 K): t = 300 K / ([solar power density] / 0.0016 J/(mm²⋅K)).
This time until the maximum safe temperature is reached (which ignores heat transfer!) is:
| f/1.2 | f/1.8 | f/2.8 | f/4 | f/5.6 | f/8 | f/11 |
| 0.044 s ≈ 0.05 s | 0.096 s ≈ 0.1 s | 0.24 s ≈ 0.25 s | 0.48 s ≈ 0.5 s | 0.92 s ≈ 1 s | 1.92 s ≈ 2 s | 3.69 s ≈ 4 s |
[Table 2]
Let us now consider different types of heat transfer/cooling.
We need to differentiate convective, radiative, and conductive cooling mechanisms.
Convective cooling is heat transfer through air molecules inside the sensor housing. For air, the convective heat coefficient is around h ≈ 10 W/(m²⋅K). For a temperature difference of 300 K (sensor to ambient air), and considering the effect on both the front- and backside of the sensor, this leads to a convective cooling power (per area) of 2⋅10 W/(m²⋅K)⋅300 K = 6 kW/m², which is negligible compared to the solar power density even for f/11 (see Table 2).
Radiative cooling is thermal radiation emitted by the sensor that heats up its surrounding housing. With an assumed emissivity of ε ≈ 0.9 and using the Stefan-Boltzmann constant, both sides radiating, and again an assumed temperature difference of 300 K (as before), we get a radiative cooling power (again, per area) of 2⋅0.9⋅5.67⋅10⁻⁸ W/(m²⋅K⁴)⋅((600 K)⁴ - (300 K)⁴) = 13.2 kW/m², which again is very small compared to the heating power density.
Lastly, the conductive cooling mechanism. It transfers heat through the sensor to neighboring areas of the sensor. We need to distinguish lateral (within the same layer) and vertical (perpendicular to the sensor) transfer. Silicon has a heat conductivity of roughly 150 W/(m⋅K), whereas the other components have vastly different values (200-400 W/(m⋅K) for wiring; <10 W/(m⋅K) for glue, polymers etc.). However, the silicon layer is going to be absorbing most of the light and therefore where the vast majority of thermal energy is going to be deposited is the silicon layer, and therefore, let's focus only on lateral conduction within this layer. As such, let us assume that the sensor has a heat conductivity of 150 W/(m⋅K), which is the value for silicon. This doesn't mean that there is no vertical heat transfer, and the other layers such as the Bayer filter etc.
will melt if the silicon layer beneath it reaches a certain temperature. Conduction strength is linearly dependent on temperature difference, meaning that stronger gradients across the sensor transport heat more quickly. Now conductivity is also inversely dependent on the conduction path distance. For very small spots (short focal lengths), the conduction path can be assumed as ~0.5 mm, whereas if the projection of the sun on the sensor is considerably larger than 1 mm², then larger path lengths come into play. Furthermore, as the adjacent areas of the sensor also heat up (which happens fairly quickly, as per Table 2), conduction path lengths do increase even for small focal lengths, as the heat basically needs to be transported "beyond the heated area", which increases. For the purpose of these calculations, let us assume that the sun is not sitting directly at an edge, but relatively centered within the frame, to allow heat conduction in all directions. Heat conduction at the very edges of the sensor is reduced by up to 50% (for edges) and 75% (for the corners), which I will ignore to keep things simple.