Determining the Sun’s Surface Temperature With iPhone

P. D. Mullen and C. N. WoodsDepartment of Physics and Astronomy,

University of Georgia, Athens, Georgia 30602

(Dated: 8 December 2015)

The principles behind blackbody radiation make it possible for one to measure the surface temper-ature of many objects based upon the distribution of emission intensity per wavelength. These sameprinciples can be even applied to measure the surface temperature of objects millions of miles away– for instance, our sun at around 93 million miles from Earth. Herein, we apply the fundamentalphysics behind blackbody radiation theory to measure the surface temperature of the sun, which isknown to be 5778 K, using a homemade spectrophotometer with optical components designed fromeveryday objects including the Complementary Metal-Oxide-Semiconductor (CMOS) type detectorof an iPhone 5S, Legos, electrical tape, and a diffraction grating with a fixed budget. Using thisdevice, we performed measurements of the sun from Athens, GA, in conjunction with the webpageapplication, Spectral Workbench. As we will discuss, several key considerations must be made re-garding the possible sources of error associated with these measurements. Firstly, calibration ofthe iPhone spectrophotometer is required to minimize error from the instrument itself: 1) we cali-brate via a known fluorescent source with characteristic mercury emission features to discern whichwavelengths of sunlight are associated with which intensities and 2) utilize two unique techniques ofcalibration (a. directing spectrophotometer directly at sunlight, b. directing spectrophotometer ata reflective surface to reduce intensity of incident light) in order to minimize effects of non-linearityin detection of the CCD within the iPhone. In addition to errors associated with the instrumentitself, we must also consider how atmospheric conditions may affect the spectrum we generate: 1)atmospheric molecules absorbing solar radiation (O2 and H2O), 2) times of measurement (intensityof solar radiation), and 3) the weather conditions (cloudy vs. sunny) that may also affect the gener-ated spectrum. At the conclusion of our construction, experiment, and data analysis, we determinethe surface temperature of the sun to be T = 6000 ± 300 K. The true surface temperature fallswithin these error bars and the percent error between the known temperature and our best estimateis a mere 3.8%.

I. Introduction: The iPhone Spectrophotometerand the Photosphere of the Sun

93 millions away from Earth, the Sun undergoes nuclearfusion at its core and reaches upwards of 15 million K.As this thermal energy radiates from the star’s core tosurface, or the photosphere, the temperature falls to 5778K. It is in the photosphere that the sun’s radiation is de-tected as sunlight. In this work, we test the fundamentalphysics (i.e. a blackbody radiation model) of the pho-tosphere by designing a simple yet intuitive spectropho-tometer using an iPhone, diffraction grating, and a fil-ter/shield to observe the emission of the Sun in effortsto measure its surface temperature. Such an instrumentis idealized in Figure 1. Discussions pertaining to theconstruction and components of this device are later ad-dressed in this report. Many concerns revolving aroundmaking measurements with this apparatus and the subse-quent application of models are addressed in this report;for instance: 1) the careful calibration of the appara-tus, 2) the times and locations for measurements of thesun’s emission, 3) fitting the observed data of a limitedrange of wavelengths to a broader scale spectrum using apowerful Python fitting program, and 4) accounting forsources of error associated with the optical components

and measurements made in this experiment.

iPhone

Diffraction Grating

Lego Spectrometer Filter/Shield

iPhone Spectrometer

Incident Light

FIG. 1: Schematic of the iPhone Spectrophotometer. Thecomponents shown above include the iPhone, diffraction grat-ing, and the Lego Spectrophotometer Tube. Not shown arethe electrical tape supports.

II. Theory: Blackbody Radiation

When heated, many bodies give off radiation. The fre-quency of radiation increases as the temperature of thebody increases. We can estimate the emission spectrumof many radiating objects by approximating/consideringthem to be blackbodies. A black body is an ideal bodythat absorbs and emits all frequencies and whose radi-

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FIG. 2: Blackbody radiation diagram for multiple blackbodytemperatures, T. λmax values are given and are precisely themaximum intensity for a given radiation curve. This figurewas obtained from http://quantumfreak.com/wp-content/

uploads/2008/09/black-body-radiation-curves.png

ation is modeled exactly by the Planck distribution asderived by Max Planck in 1900. To be explicit, in thiswork, the photosphere of the sun is modeled as a black-body. Blackbody radiation perplexed physicists prior tothe beginning of the 20th century as their derivations in-corporating classical physics reported that intensity di-verged to infinity at the ultraviolet wavelengths (i.e. theultraviolet catastrophe). In 1900, Max Planck incorpo-rated his idea of quantized energy and was able to derivewhat is presently the most commonly used distributionfor blackbody radiation and opened the door to the fieldof quantum physics. This distribution is given as

ρλ(T )dλ =8πhc

λ5dλ

ehc

λkBT − 1(1)

This distribution is plotted in Figure (2) for several dif-ferent objects of different surface temperatures, T. Also,Figure (2) depicts that the radiation corresponds to allwavelengths of emitted light. For reasons that will laterbe discussed, for temperatures around 5000-7000 K, theλmax falls within the visible portion of the emission spec-trum. One can explictly find the maximum wavelengthof the blackbody radiation by differentiating equation (1)and setting it equal to zero. Here, we can determine thatthe λmax associated with the emission spectrum is givenas

λmaxT = 2.90 × 10−3 m ·K (2)

Thus, if we know λmax, we can determine the surfacetemperature of the radiating object from equation (2).This is how we will measure the surface temperature ofthe sun. As discussed in later sections, we will attemptto find λmax through a series of observations with theiPhone spectrophotometer, apply a powerful Python fit-ting tool to extract the entire spectrum and find thewavelength, λmax, which maximizes the extrapolatedblackbody radiation curve.

FIG. 3: Execution of iPhone Spectrophotometer Construc-tion.

III. The Apparatus

A. Construction

We will be using the rear camera of an iPhone 5S asthe detecting device in our experimental set up. Thecamera is a custom Sony Exmor RS device, which isa Complementary Metal-Oxide-Semiconductor (CMOS)type detector. The iPhone is well suited as a ready touse spectroscopic device due to the presence of webpageapplications which can be used to extract/generate ob-servational spectra. As shown, in Figure 1, to build theidealized spectrophotometer, the main optical devices re-quired were an iPhone 5s, a diffraction grating, and aLego Spectrophotometer Filter/Shield. This constructedLego filter/shield allows for a channel of light from a de-sired source, in this case, the sun (photosphere), to bedirected to our detector while hopefully limiting as muchexternal interference as possible. To this filter/shield,a diffraction grating is attached. The diffraction grat-ing will separate out various wavelengths of the incominglight so that the iPhone 5S camera can generate a spec-tral picture in which the webpage application, SpectralWorkbench, can analyze.

Until now, we have presented the idealized spectropho-tometer and its components and their purpose in themeasuring the surface temperature of the Sun. However,the actual construction of the device required special ad-ditions to secure such devices. Therefore, in addition tothe aforementioned, the spectrophotometer also requiredan iPhone 5s case, black tape, super glue, and a drillfor assembly. First, holes were drilled through the cen-ter of Lego pieces and then the pieces were attached sothat the hole went all the way through the construction

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– thus generating the filter/shield. Over the hole on thecase, the diffraction grating was taped so that only athin slit of grating was visible. The grating was tapedso that the lines of the grating were perpendicular to thelength of the case. The tape was placed parallel to thelines of the grating. The hole on the bottom side of theLego construction was also taped over so that there wasonly about 1 mm of space between the tape. The Legoconstruction was then glued over the diffraction gratingtaped to the phone case so that both slits were lined up.Additional tape was used to secure the Lego constructionin place while the glue set. Black was used for all partsof the construction so that effects of scattering from theconstruction were limited to ensure as pure of a spectrumas possible. All edges were also taped so that no ambientlight could get into the optical piece. The only materialspurchased were the tape, glue and phone case. A pictureof our execution of constructing the iPhone spectropho-tometer is given in Figure 3. The total cost was $14.67.For itemized expenditures, see the cost section of thisreport.

B. Calibration

It would be useless to use our device without cor-rectly calibrating it for spectral response. To this end,we required for calibration to take a leading role in thisproject. The CCD cameras in typical cell phones gen-erally have a wavelength response in the range of 350 -1000 nm. This means that our cell phone detector willbe able to detect the peak solar wavelength as it shouldfall in this range. The CCD cameras also show non-linear responses in regards to intensity. To correct fornon-linearity in the intensity measurements, we variedthe intensity of the light in our observations to detectand correct for non-linearity in the intensity response ofthe CCD detectors. To be more specific, two techniqueswere applied in observations: the first was generating a“dark spectra”, obtained from fluorescent source in whicha plastic covering lowered the intensity of the source. Thesecond was a “bright spectra”, obtained by removing theplastic covering and using the unblocked intensity of thelight. By comparing the “bright spectra” analysis to the“dark spectra” analysis, we minimize the effects of theintensity response of the CCD detectors and implementsuch differences in the two methods into our uncertaintyin observation. No matter which technique is appliedto minimize the effects of the non-linear response of theCCD camera, the generated spectral image remains to becalibrated (i.e. the device must be calibrated so that thewavelengths associated with the intensities of the sourcecan be determined). In order to obtain the wavelengthsof the spectrum, the site uses the spectrum taken withthe device from a fluorescent light bulb to do this cal-ibration. Two peaks in the fluorescent bulb spectrumare identified by the user which are typical of the spec-trum. These peaks correspond to mercury emission fea-

FIG. 4: Calibration Technique: Identification of Mer-cury Peaks in Fluorescent light spectrum. This fig-ure was obtained from https://publiclab.org/wiki/

spectral-workbench-calibration

FIG. 5: Raw Data: Spectral Image Prior to Processing

tures from the fluorescent light source as seen in Figure4 which was gathered from the Spectral Workbench cali-bration instructions.

C. Detection Methodology and ObservationalChallenges

For the purposes of measuring the solar spectrum, weused a fairly simple methodology. To obtain the spectra,the tube of the device was directed at the light sourcein question and then the angle was changed until a spec-trum appeared on the image. A picture was then taken.There was generally no diffraction pattern when the de-vice was directed straight at the source. The diffractionpattern is due to reflection of the source light inside thetube of the spectrophotometer. The resulting image givesthe raw spectral data to be implemented into the Spec-tral Workbench application. An example of such a rawimage is given in Figure 5. The image must croppedso that only the spectrum was present and oriented sothat the violet region was on the left and the red regionwas on the right. After calibration (as mentioned in theprevious section), the capabilities of the Spectral Work-bench site were then used to obtain a spectrum from thecolored image. These measurements were taken over thecourse of one day from 10 a.m. to 4 p.m. once an hourto correct for atmospheric changes and intensity changesthroughout the day. The measurements were made inAthens, GA, on a clear sunny day. To give a qualita-tive and representative picture of these conditions, Fig-ure 6 depicts the sky on observation day as viewed atopUGA’s journalism building. As viewed in the picture,we chose a day that would hopefully eliminate any at-

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FIG. 6: Observation Day Conditions: Viewed Atop UGA’sJournalism Building

FIG. 7: Peculiarities of observed radiation curves due to ab-sorbance of radiation by atmospheric molecules O2, H2O, andCO2 and dependence of elevation at which measurement istaken. Picture obtained from Dr. Yiping Zhao’s Chapter 2UGA PHYS 3330 Lecture Notes.

mospheric peculiarities of the region that could hinderour efforts. However, more generally, it is well knownthat many molecules within Earth’s atmosphere absorblight thus leading to unusual features in the blackbodyradiation curve. For instance, Figure 7 depicts the ab-sorbance of radiation by O2, H2O, and CO2 in Earth’satmosphere. Further, the figure shows how even the el-evation at which the measurement is taken could affectthe observed radiation curve.

IV. Model Fitting – Python Routines andStatistical Analysis

In reference to Figure 2, it is imperative that one under-stands that the iPhone spectrophotometer is only capa-ble of observing electromagnetic radiation in the visiblepart of the emission spectrum. A series of data points

can be extracted from the Spectral Workbench as dis-cussed in the detection methodology section. In an effortto determine the λmax of the emission spectrum and tomodel the curve with a blackbody radiation distribution,we apply a powerful Python program, named Helios,that takes the observational data from the iPhone spec-trophotometer uses it to fit to the model distribution asgiven in equation (1). From here, we can deduce the tem-perature associated with the emitting object directly, orwe can find the maximum wavelength associated withour extrapolated distribution and find the correspond-ing surface temperature using equation (2). The pythonfitting routine is robust and fast. A series of 14 observa-tional data sets are analyzed in a fraction of seconds whileanalyzing quantitative statistics such as the chi-squaredvalue, reduced chi-squared value, degrees of freedom, p-values, percent errors, and percent differences all whiletracking the errors associated with the measurements andmaking 28 figures corresponding to fitting each set of dataand giving residual analysis. Thus, by accounting for er-ror and applying such powerful modeling tools, we cangive the the sun’s surface temperature, with uncertainty,and statistical parameters supporting our results.

V. Results and Discussion

Helios was applied to 14 different sets of observationaldata spanning an observation period from 10 a.m. and 4p.m. – 7 using our defined “bright spectra” and 7 using“dark spectra”. The output of the streamlined programset is 28 figures comprised of model fits and residual anal-ysis. We will focus on 4 of these graphs. Figures 8 depictsthe observational data (green points) and the correspond-ing fit to a blackbody radiation distribution (red curve)utilizing the “bright spectra” data. It is very importantto note that each of these spectra have been normal-ized to the intensity occurring at λmax. This has beenperformed for ease of comparing spectra and is often acustomary mode of analysis for emission spectra. Qual-itatively, we can see that there is excellent agreementbetween the theory and observations. More powerfully,statistical analyses also reveal an excellent fit with a re-duced chi-squared value of 0.004549 and a p-value veryclose to ˜1.0. We see further that the the theoreticalcurve mostly falls within the error bars of the intensitymeasurements. These error bars were determined by con-sidering the change in intensities of the same wavelengthfor spectra taken in rapid succession and is equal to 0.05.From the fits and data, a λmax value of 493 ± 2 nm wasfound which corresponds to a temperature of 5880 ± 20K via Equation 2. The origination of uncertainty comesfrom the 2 nm uncertainty in identifying the location ofthe mercury emission peak in the calibration procedure.Figure 9 gives a residual analysis for this fit. We see thatthe residuals float around 0.00 but there is a particularlyinteresting feature occurring around ˜640 nm. Althoughour main goal for this work was to identify the sun’s sur-face temperature, we are very excited by this peak as we

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Observational DataPlanck Blackbody Curve Fit

FIG. 8: Model Fit and Observational Data from Bright Cali-bration Method.

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FIG. 9: Residual Analysis Bright Calibration Method.

know this is around the location for atmospheric O2 vi-brational absorption (630 nm and 690 nm) as depictedin Figure 7 which clearly explains the dip in intensitiesand increase in the residuals about this wavelength.

Figure 10 depicts the observational data (again, green)and the corresponding fit to the blackbody radiation dis-tribution function (red curve). Again, we see excellentqualitative agreement and an excellent chi-squared valueof 0.004321 an a p-value again very close to ˜1.0. Fromthe fits an observational data, a λmax value of 497 ± 2nm was determined which gives a temperature of 5830 ±20 K. Figure 11 gives the residual analysis which floatsaround 0.00 but again shows the very interesting (butless distinct) peak that we attribute to O2 absorption inEarth’s atmosphere. Utilizing the results from every setof observational data, “bright spectra” and “dark spec-

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Sun Spectrum: Intensity vs. Wavelength

Observational DataPlanck Blackbody Curve Fit

FIG. 10: Model Fit and Observational Data from “Dark”Calibration Method.

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tra”, Helios averages each temperature while consider-ing all uncertainties so that we can report our best es-timate and uncertainty for the sun’s temperature whichwe find to be T = 6000 ± 300 K. The true value ofthe photosphere’s temperature is 5778 K. Therefore, wefeel successful in reaching our project goal as we haveachieved a 3.8% percent error in this known value whilealso having been able to identify key absorption featuresin Earth’s atmosphere.

VI. Sources of Error

As this is a rather simple device, there are quite a fewsources of error in this experiment. One source of errorcomes from the tube of the device. The Legos themselvesare able to scatter the incoming light from the source and

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this will affect the spectrum. Diffraction caused by theLegos will alter the light that is incident on the diffrac-tion grating. Absorbance by the Legos might also lowerthe intensity of the light at certain wavelengths that hitsthe diffraction grating. We used black Legos to minimizethe effects of the absorbance, as the Legos should absorbmore in the infrared than the visible spectrum. If oneused colored Legos, this would have to be controlled for,as they would absorb some wavelengths in visible spec-trum, which is our region of interest. Another source oferror comes from the orientation of the spectrum. Thespectrum was oriented so that the rainbow pattern wasas horizontal as possible. Deviations from a spectrumaligned on axis would cause the program used to gen-erate the spectrum from the image to possibly over orundercount the intensity of wavelengths. As mentionedbefore, the spectral response of the CCD camera used tocapture the image is also a source of error. The cam-era of the iPhone 5S is a custom Sony Exmor RS device,which is a Complementary Metal-Oxide-Semiconductor(CMOS) type device. These devices seem to generally re-spond more to higher wavelengths, reaching a maximumresponse somewhere around 700 - 1000 nm. This meansthat our spectra will be skewed, with higher wavelengthsresponding more. This should not cause too much of anissue as the solar spectrum reaches a maximum intensityat lower wavelengths in the range of 500 nm. It will how-ever effect the fit some, showing a broader peak than itactually is. The best way to correct for this would be toutilize a detector which shows nearly constant responsein the visible region. As the device is rather simple, thespectral resolution leaves a lot to be desired. This limita-tion became very apparent in the calibration procedure.Some of the peaks were ill defined which made it accu-rately to assess which peaks need to be selected for thecalibration procedure. A sharper spectrum would makethe calibration much more accurate and therefore helpto nail down the wavelengths in the solar spectra. As weare interested in finding the wavelength of maximum in-tensity this is most likely the largest source of error. Theuncertainty on our x-axis was determined to be about2 nm. This value was obtained from the width of thewidest peak used in the calibration procedure. It mayseem as if the FWHM value would be more appropriateand not the base width, but as there is error inherent inthe orientation of the spectra we found it better to pro-ceed with caution and implement the larger value intoour uncertainty.

VII. Conclusion

In conclusion, we have carefully implemented the prin-ciples behind blackbody radiation and Planck’s distribu-tion to cleverly identify a means of estimating the surfacetemperature of many emitting objects – but of particularinterest, our own Sun’s photosphere that is over 93 mil-

lion miles away. By adhering to a very cost-effective andminimal budget, we have used everyday items, such as aniPhone, Legos, and tape and specialized items, such as adiffraction grating, and a webpage application, SpectralWorkbench, develop a iPhone spectrophotometer thatcan be precisely calibrated to deduce wavelengths and in-tensities of visible light emission from the Sun. Using thisobservational data, we can extrapolate the blackbody ra-diation curve in its entirety and ultimately determine thesurface temperature of the sun by finding the wavelengththat maximizes intensity. Associated with all of thesemeasurements are a variety of sources of errors rangingfrom the simplicity of the spectrophotometer design (andiPhone detector), the atmospheric conditions (i.e. O2

and H2O absorbance in atmosphere), errors associatedwith the Spectral Workbench application, errors in cali-bration techniques, and finally, error in fitting routines.All of these errors have been carefully tracked, reported,and included in the final reporting of the surface tem-perature of the sun and the uncertainty in measurementwhich we ultimately find to be T = 6000 ± 300 K whichis a mere 3.8% in difference from the known value. Wehave also, unexpectedly, identified atmospheric moleculeabsorptions in our spectra around ˜640 nm which givefurther merit to our spectrophotometer’s abilities to ac-curately measure blackbody radiation. Future endeavorsin this work involve applying Helios to a variety of othersystems that can be modeled by a blackbody radiationcurve to see if our calibration, observational, and analyt-ical techniques again yield excellent fits.

VIII. Costs

-iPhone 5S Case: $4.88-Tape: $4.86-Glue: $3.97

Sum (w/ tax): $14.67

IX. References

-Darmon, Arnaud. (2009). Spectral Response of SiliconImage Sensors. Aphesa. www.aphesa.com.

-McQuarrie, Donald A.; Simon, John D. PhysicalChemistry: A Molecular Approach. University ScienceBooks (1997).

-Phillips, K. J. H. (1995). Guide to the Sun. CambridgeUniversity Press. pp. 4753. ISBN 978-0-521-39788-9.

-Zhao, Y. PHYS 3330: Optics (2012). Chapter 2:Blackbody Radiation. Department of Physics andAstronomy, University of Georgia.