Teacher s guide CESAR Science Case Rtatin perid f the Sun and the sunspt activity The students can use different ways during the labratry and there are diverse methds t use. Material that is necessary during the labratry CESAR Astrnmical wrd list CESAR Bklet CESAR Frmula sheet CESAR Student s guide CESAR images f the Sun Matlab, Octave (ptinal) The sftware fr this Science Case Paper, pencil, ruler, calculatr, prtractr Intrductin The tasks that the students need t cmplete are the fllwing: The first ne is t get a set f 3-7 images f the Sun separated by 24h (at least) in the time f acquisitin. The secnd task fr the students is t calculate the heligraphic crdinates f any f the sunspts. Aligning these images with a slar grid is the easiest way but nt the mst precise. Using these crdinates, they will be able t estimate the rtatin perid f the Sun. An extra task f this case is t estimate the size f a sunspt and cmpare it with the size f the Earth. T finish with, and t see if they understand the sunspt activity, they shuld be able t estimate where we are in the 11-years slar cycle and try t predict when the next maximum and minimum activity will ccur. This is pssible by pltting the given data, either by hand r sftware. Fr the first task, there are tw pssible slutins. If the rbtic CESAR slar telescpe is available fr bservatin and image capturing, yu can dwnlad real-time images frm the CESAR website. If the CESAR slar telescpe is nt available r if yu dn t want t use it, the ther ptin is t dwnlad a set f preselected images frm the CESAR website fr this Science Case and use it. Obviusly part f the educatinal purpse is t apprach prfessinal telescpes t the students s they culd learn hw prfessinal bservatries actually wrks. Mrever they wuld be able t see hw remtely a telescpe mves as they want (pinting the Sun in this Science Case). It s nt necessary t get a whle set f 7 images, s this Science Case culd be dne by using tw (preferable three) images separated three hurs. It culd happen that the day f bservatin n sunspts are visible. In that case, it s impssible t d this labratry. Fr this reasn yu culd check if there are sunspts available befre at http://sh.nascm.nasa.gv/sunspts/ At this pint it is suppsed that yu already have the set f images. First, start by handing ut all the material. Give them the bklet in advance s that they are well prepared. It s very imprtant fr them t understand what sunspts are, hw d they change and hw d they mve during the 11-years cycle. At the CESAR website fr this Science Case, yu have all the sftware that is needed fr the labratry. Hwever, the free sftware named GIMP is anther gd chice fr the tasks that may require image prcessing. It is dwnladable at http://gimp.cm. T help with the pltting f the 11-years slar cycle we prvide sftware t, but yu culd instead use sftware that allws yu t plt the graph. 1
Sunspts tracking Chsing the sunspts frm the set f images As mentined in the labratry descriptin, yur students need t get the images and its dates fr the bservatins. They need t name sme f the sunspts t keep them n track. What name it des nt matter, but each spt shuld have its individual name. Furthermre it s imprtant t chse thse spts that are clser t the centre f the Sun. It is preferable because it s easier t appreciate the displacement f a sunspt n the images if it is near t the centre and cnsequently it s easier t d the measures. After taking the images with the CESAR telescpe, decide if it s wrth t prcess the images r nt. It shuld nt be necessary since the quality prbably will be gd enugh. The sunspts must be clearly visible and nt blurred. If yu cnsider that sunspts are nt clearly visible, try t increase the cntrast with an image prcessing prgram in rder t distinguish them. A very imprtant thing that has t be mentined is that the images have t be aligned in E-W directin in the slar disk. This is the first requirement t determinate the rientatin f the Sun and its heligraphic crdinates. The CESAR images will already been aligned but if yu have cnsidered t d this Science Case using yur wn telescpe with an equatrial munt a trick t get the E-W directin is t take pictures while mving in Right-Ascensin. If yu take tw pictures fr example and superpse them int a single ne, draw a line that cnnect the same sunspt f the tw frames. That is the E-W directin, and yu can nw rtate the image t see the E-W directin parallel t the mark frame as shwed belw. Figure 1: Image shwing the E-W directin where tw pictures taken at tw separated dates where superpsed. Credit: CESAR Until nw, we have a set f images captured at different times and riented n E-W directin. Nw we are ging t determinate the heligraphic crdinates fr the sunspts. 2
Heligraphic crdinates Determine the heligraphic crdinates f the sunspts The heligraphic crdinates are the crdinates f the Sun. As well as a place in the Earth has a lngitude and a latitude in the Sun is exactly the same. A sunspt has a lngitude and latitude n its surface t. Its latitude is divided int 90 degrees nrth t the equatr, and 90 degrees suth t the equatr. Its lngitude is divided int 360 degrees t the West in the directin f the slar rtatin. The Sun as the Earth has a place frm which t start cunting the lngitude degrees. As well as the Earth has a rtatinal axis, the Sun t. It s well knwn that the Earth rtatinal axis is tilted 23,5 degrees while the Sun s rtatinal axis is tilted 7 degrees. If bth Earth and Sun rtatinal axis were parallel between them and perpendicular t the ecliptic, then the E-W directin that we mentined befre wuld be the equatr f the Sun. Figure 2: Inclinatin f the slar axis alng a year viewed frm Earth. Credit CESAR What we are ging t d nw is t calculate the heligraphic crdinates f the sunspts using the crdinates f the sunspts in the E-W and N-S crdinate system. In rder t d this, students first have t measure the psitin f the sunspts frm the center f the Sun. The center f the Sun has t be lcated because is the center f ur reference system. Students have t create a table like this: Sunspt X crdinate Y crdinate R m R 1 356 pixels 232 pixels 425 pixels 3115 pixels Where R % is X ' + Y ' and R is the diameter f the Sun that they have t measure t (nly nce). We need nw t calculate ther values using the X and Y crdinates f the sunspts. These values are the psitin angle f the sunspt measured frm N-S directin (that we call P % ) and the angle between the sunspts with the visual (that we call ρ). Their frmulas t be calculated are: 3
P % = arc tg( X Y ) (1) and ρ = arc sin (R %/R) α/2(r % /R) (2) Where α is de angular diameter f the Sun (n degrees), that yu culd get n internet r via sftware (the free sftware Stellarium is a gd chice) fr the day f the bservatin. Yu culd als use the value f 0.5244 fr summer, 0.5422 fr winter and 0.5333 fr autumn/spring in the Nrth hemisphere. As well as in the first table, the students have t nte the values f P % and ρ fr each sunspt in a table: Sunspt P m ( ) ρ ( ) 1 56,9 7,8 As we said, the Sun s axis is tilt s we see different angles with the N-S directin alng a year. Furthermre the center f the heligraphic crdinate system is nt the place where E-W and N-S lines crss each ther. Fr all f this, we need t take sme values which are: P: Is the psitin angle f the slar rtatin axis measured frm the N-S directin. Psitive when it s t the East and negative when it s t the West. B? : Heligraphic latitude f the slar disk center. L? : Heligraphic lngitude f the slar disk center. T make this Science Case easier, we culd take these values frm the Ephemeris. There are sme websites such as: http://alp-astrnmy.rg/slarblg/wp-cntent/uplads/ephems/slar_ephemeris_2016.pdf http://bass2000.bspm.fr/ephem.php Here there is a picture shwing all these values. Figure 3: Picture shwing all mentined values. Credit: CESAR 4
At this pint, the students have t be able t calculate the heligraphic crdinates f any f the sunspts. It wuld be great if the students calculate the crdinates fr at least tw sunspts. Fr this here there are the equatins: sen B = sen B0 cs ρ + cs B0 sen ρ cs P Pm (3) sen(l L0) = sen(p Pm)sen(ρ)cs (B) (4) Where B is the latitude and L the lngitude. Nte that all f the values that have t be used in the frmula are expressed in degrees as well as the results. T sum up, the mst imprtant part is t write dwn all the data n a paper. The students shuld create a final table as the ne belw: Date/time Sunspt Lngitude( ) Latitude( ) Observatins 17/11/14 12:00 1 242,813-18,209 This is imprtant: as sme f the frmulas are a bit difficult depending n the knwledge f the students, the sftware at the website simply ask the students t d sme measurements and then autmatically calculate the heligraphic crdinates. Grid alignment Align the images with a grid f heligraphic crdinates Nw it s pssible t make sme interesting calculatins but befre that we are ging t use a grid that will be essential if we want t estimate the heligraphic crdinates f a sunspt withut ding all the measurements that we explained befre. The first step fr this is t dwnlad the exact grid fr the day when the images were taken. Here is the website fr this: http://bass2000.bspm.fr/ephem.php At the website yu just have t select the date and time and select grid t cmpute. Once it has been dwnladed we need t superpse it t the image. There are tw ways t d this, the first ne is t print bth (the image and the grid) and draw the sunspt ver the grid. The ther ptin is t superpse the grid using sftware. With this Science Case we prvide sftware t d that. In bth cases, is essential t rtate the image the angle P t align the image with the nrth slar ple. Nw it s very easy t estimate the psitin f any f the sunspts. We are als able t determinate the size (in terms f heligraphic crdinates) f a filament (if fr example a h-alpha filter were used) r the size f a grup f sunspts as shwed in the Science Case: Sun s differential rtatin. The picture result f superpsing the image and the grid shuld be similar t the ne belw: 5
Figure 4: Heligraphic crdinates superpsed t an image f the Sun. Credit: CESAR/LEISA Rtatin perid Calculate the Sun s rtatin perid using sunspts T determinate the Sun s rtatin perid tw images at different times are required. The images have t be at least three r mre hurs separated ne frm the ther. In fact the best idea is t use tw images frm tw different days t d the measurements easier. As an example, the figure 1 shws tw pictures f the sunspts three days spaced. That wuld be the ideal. Nw tw different ways t calculate the rtatin perid are ging t be presented. The first ne requires sme knwledge abut sidereal perid whereas the ther requires a gd understanding f trignmetric. In bth cases they need t write dwn the heligraphic latitude f the same sunspt fr bth images. The latitude is nt required this time but the students have t repeat the calculatins mentined befre t get the lngitude. S the first task fr the students is t fill this table: Date/time Lngitude ( ) ΔL( ) Where L is the difference f lngitudes f the sunspt in degrees. Yu may cnsider using mre images t track the sunspt. In that case, students can take the mean f the results t ensure a gd result. 6
The next step fr the students is t understand the difference between the sidereal rtatin perid (that we call P) and the syndic rtatin perid (that we call S). The first ne is the time that a sunspt needs t cmplete a 360 degrees rtatin whereas the secnd ne is the time that a sunspt needs t rtate t the same apparent psitin viewed frm Earth. Figure 5: On the left, the representatin f a sidereal rtatin f a sunspt. On the right, the representatin f a syndic rtatin which takes mre time t be cmpleted (indicated by the yellw arrw. Credit: CESAR T calculate the syndic rtatin perid the students had t realize that it s as easy as calculate the rate between time and degrees. This is achieved by dividing the time between the tw images (Δt) with the change f lngitude (ΔL). Then they have t calculate hw much time it takes t cmplete a 360 rtatin. The equatin that the students need t use t calculate S is then: S = 360 t L (5) The value f S shuld be expressed in days, hurs, minutes and secnds, and it shuld be clse t the exact value f 26 days and 14 hurs. Finally, t calculate the sidereal rtatin perid f the Sun which is the main target f this sectin, we need t remve the effect f the Earth s translatin. P = (S 365,25)/(S + 365.25) (6) This is the real rtatin perid f the Sun, and the students shuld get an apprximate value f 27d 6h 36. The ther prcedure that we are ging t use is based n sme trignmetric frmulas. A calculatin f this rate can be achieved like this (there may be several different ways this is just ne f them). As we have already seen, the mvement f a sunspt can be used t btain the rate f the Sun s rtatin. One methd t calculate the rtatin perid f the Sun is t imagine the sunspts seen frm the tp f the 7
Sun. By ding this, the angle f slar latitude and lngitude can be calculated. Here is an image f hw it might lk like: Figure 6: The upper image is the Sun frm abve. The bttm left image is the Sun viewed frm Earth, and the bttm right image is the Sun frm the center. Credit: CESAR In the figure 6, α is the slar latitude, β 1 is the slar lngitude angle f the spt at first time while β 2 is the angle f the spt at time tw. The first ne is measured frm the edge f the Sun. As the student may knw, the sunspts are always lcated at the same latitude measured frm the equatr f the Sun (r frm the ple equivalently). Alng the line f latitude, the student can identify fur lcatins alng that crrespnding latitude that are distinguished by their x-crdinates. Here is an example f a crdinate table that the student may cme up with and that is helpful. X 1 X 2 X 3 X 4 The sunspt n the left edge Sunspt in the first image Sunspt n the secnd image The sunspt n the right edge We are nw at the pint where the β 1 angle needs t be figured ut, but als β 2 frm the left edge f the secnd image f the spt. The tw angles have a difference: β = β2 β1 (7) 8
This is the Sun s angle f rtatin between the times f the tw images. We first need t find the Sun s semi-diameter (als knwn as the radius) in pixels (depending n what methd they chse) at that latitude. We can call it φ. Using the figure belw it is bvius with trignmetric relatins that: Figure 7: Trignmetric explanatin. Credit: CESAR φ = x S x T 2 (8) If we take a lk at the first image, we can cnclude that the angle β 1 satisfies the trignmetric relatin: cs β T = T V φ x ' + x T = WXYWZ ] [ [^] Z WXYWZ [ giving: β T = cs _T `_(] [_] Z ) ` (9) Here, x 1 and x 2 are the crdinates f the edge t the left f the Sun and f the sunspt measured at that same latitude. T get the secnd angle, β 2, we need a mre cmplicated evaluatin. Take a lk at the figure. It can be seen frm it (by trignmetric relatins) that: cs π β ' = cs (β ' ) = bcdefghi jkcg lmnighpjg = ] q_ V V giving: β ' = cs _T ] q_ V V (10) β = β2 β1 = cs _T ] q_ V V cs _T `_(] [_] Z ) ` (11) As we saw in the previus methd, the sunspt s time f rtatin at this height r latitude is the time it wuld take fr it t g arund the entire 360 f lngitude at the slar latitude. These times can be called t 1 and t 2, and nce they are determined yu can calculate the time interval t = t ' t T (in days): The lngitudinal angle s rate f change that is measured in degrees/day is: w = fjyz WqY x x i fj YZ yy(w [YWZ) y i [ _ i Z (12) And the sunspt s rtatin perid T expressed in days is (nte that is similar t the equatin five): 9
T = 360 t β (13) We knw that the Sun is a gaseus bdy (rather than a slid ne). Its rate f rtatin depends t sme degree n its latitude. It actually rtates faster at the equatr and mre slwly at higher latitudes as it s develped in the Differential rtatin f the Sun and the Chrmsphere Science Case. These are frmulas that can be used. Of curse, the student is free t chse what methd t use. Size f a sunspt Get the size f a sunspt r a grup f then and then cmpare it with the Earth size It was mentined in the Science Case that it is pssible t calculate the sunspts sizes by using the images. Let the students lk up the values fr the diameter f the Sun and the Earth in kilmetres, and the diameter f the Sun and a spt n their images in pixels (this extra task is als included in sftware fr this Science Case where they culd measure this sizes n pixels). If they have the same units, they can crss multiply t get the magnitude in size f the sunspt in km. Here is ne example: The size f the sunspt: 21 pixels The size f the Sun n the drawing: 936 pixels Average diameter f the Sun: 1.391.000 km Diameter f Earth: 12.742 km T measure the size f a sunspt they culd measure diameter f the umbra r the diameter f the umbra and the penumbra t, that is larger. They culd als measure the size f a grup f sunspts. The simple relatinship t calculate the value is the fllwing: Measured size f the sunspt/measured size f the Sun = Real size f sunspt/real size f the Sun (6) Fr the example we get: 21 1.391.000 Real size f sunspt = = 31.208 km 936 And finally, hw many Earth diameters can fit in a sunspt with the calculated size? The simplest way t cmpute it is t divide the size f the sunspt by the Earth s diameter Size f sunspt in Earths = Real size f sunspt/size f Earth (7) Numerically: Size f sunspt (in Earths) = 31.208/12.742 = 2. 45 Earths! The Earth is apprximately 110 times smaller than the Sun s the students have t realize abut this. Slar Cycle 10
Predictins abut the slar cycle using different methds The next exercise is t use the ld sunspt data that is prvided and let the students plt it t see if they can predict if we are in a slar maximum r minimum r in the middle f bth. Of curse, the student can plt the cllected data n a paper by hand but is preferable using sftware. It is basically up t them, but yu as a teacher yu shuld maybe g thrugh simple Matlab (r anther language) cding just t get an apprach t sme prgramming. Pwer pint is als a gd sftware t use when data needs t be pltted. The website http://slarscience.msfc.nasa.gv/greenwch/spt_num.txt has data frm January 1949 up t nw. They can basically chse whatever year they want, r use the examples we prvide. If they want t pick different years, they need t calculate the average SSN (Sun Spt Number) fr each year. This can be dne by adding the SSN fr each mnth in each year, and then divide it by 12 mnths. After fllwing these steps, they shuld get a graph similar t this ne belw. By lking at it, they shuld cnclude if we are in a maximum r a minimum. It is bvius in the plt that we are n a slar maximum. Slar cycle SSN 250 200 150 100 50 0 1940 1950 1960 1970 1980 1990 2000 2010 2020 There is anther methd t estimate where we are in the slar cycle. It is based n the Maunder Butterfly Diagram explained in the Bklet fr this Science Case. Based n it, what we need t use is the latitude values f the sunspts. With a range between 5 and 35 degrees, the clser a sunspt is t the equatr, the clsest t the minimum we are. If the cycle takes 11 years n average t be cmpleted, an easy prprtin will give the students an idea f where we are n in. Let s take an example: imagine that the bservatin day there are tw sunspts with different latitudes. Then the students have t create a table like this: Sunspt number Latitude f the sunspt 1 15 N 2-19 S Either if the sunspt is the nrth r suth, the abslute value is the ne that have t be cnsidered. S the students have t take the mean f the latitudes (17 in this case). With a table that shws the variance f the latitude with the percent f cmpleted cycle it easy t check that 17 degrees equals t 60% f the slar cycle. 11
Sunspt latitude Cycle's year Cycle's percept Estimate date 35 0.00 0.00 Jan 2008 - Mid May 2008 34 0.37 3.33 Mid May 2008 - Sep 2008 33 0.73 6.67 Sep 2008 - Mid Feb 2009 32 1.10 10.00 Mid Feb 2009 - Jun 2009 31 1.47 13.33 Jun 2009 - Mid Nv 2009 30 1.83 16.67 Mid Nv 2009 - Mar 2010 29 2.20 20.00 Mar 2010 - Mid Aug 2010 28 2.57 23.33 Mid Aug 2010 - Dec 2010 27 2.93 26.67 Jan 2011 - Mid May 2011 26 3.30 30.00 Mid May 2011 - Sep 2011 25 3.67 33.33 Sep 2011 - Mid Feb 2012 24 4.03 36.67 Mid Feb 2012 - Jun 2012 23 4.40 40.00 Jun 2012 - Mid Nv 2012 22 4.77 43.33 Mid Nv 2012 - Mar 2013 21 5.13 46.67 Mar 2013 - Mid Aug 2013 20 5.50 50.00 Mid Aug 2013 - Dec 2013 19 5.87 53.33 Jan 2014 - Mid May 2014 18 6.23 56.67 Mid May 2014 - Sep 2014 17 6.60 60.00 Sep 2014 - Mid Feb 2015 16 6.97 63.33 Mid Feb 2015 - Jun 2015 15 7.33 66.67 Jun 2015 - Mid Nv 2015 14 7.70 70.00 Mid Nv 2015 - Mar 2016 13 8.07 73.33 Mar 2016 - Mid Aug 2016 12 8.43 76.67 Mid Aug 2016 - Dec 2016 11 8.80 80.00 Jan 2017 - Mid May 2017 10 9.17 83.33 Mid May 2017 - Sep 2017 9 9.53 86.67 Sep 2017 - Mid Feb 2018 8 9.90 90.00 Mid Feb 2018 - Jun 2018 7 10.27 93.33 Jun 2018 - Mid Nv 2018 6 10.63 96.67 Mid Nv 2018 - Mar 2019 5 11.00 100.00 Mar 2019 - Mid Aug 2019 Be careful because the last clumn shws an estimatin f the date, but it has been cnsidered that we are in an 11 years slar cycle and that s nt always true. Sme histrical cycles tk 14 years t be cmpleted while thers tk just 9 years. Fr this reasn the last clumn shws if we are clse t an 11 years slar cycle r nt. Finally ne last methd t estimate if we are clse t the maximum r minimum is t determinate the Wlf number. The Wlf number is calculated by cunting the number f individual spts (s) and the number f sunspt grups (g) and using a factr that depends n the instrumentatin that we use (k). Fr this exercise we are ging t take k = 1. The frmula is the next: W = k 10g + s (14) 12
T cunt the number f sunspts and grups yu culd either use the images and let the students cunt them n a paper r use the sftware prvided t calculate the Wlf number by clicking the sunspts and grups n the upladed image. The Wlf range f values is between 0-10 fr minimum activity and 140-170 fr the maximum. This methd smetimes fails because we can nly bserve half f the slar disk s it culd happen that all f the sunspts appeared in the nn-bservable face f the Sun. Furthermre the images that we prvide cannt be cmpared with the images f space slar telescpes such as SOHO. Fr this reasn sme f the sunspts wn t be visible and the estimated Wlf number that the students will give t yu will be prbably belw the fficial number fr that day that yu culd take frm the website: http://www.astrsurf.cm/bslar/wlfnumber.html When all f these three methds were dne, a glbal idea f where in the slar cycle we are shuld be knwn. Have a discussin sessin with the students and see if they have understd the thery behind slar activities. If the thery is nt cmpletely understd, make sure t explain and answer the questins. 13