Get To Know Your Stars
Adapted from Kalée Tock and Ryan Caputo
Determining the Physical Relationship Between Your Stars
As explained in the introductory assignment, our goal for double stars is to figure out whether they have any physical connection to each other, and if so, what kind of connection that is. If the stars are moving together through space, they might have been born together, and ejected from the same gas cloud at a similar time and in a similar direction. These stars are called "physical systems" because they are physically related to each other. They are interesting because they give us information about the stellar nurseries from which they originated. However, even more interesting (for reasons discussed in the introductory assignment) is the subcategory of physical systems in which the two stars are gravitationally bound to each other. These are called "binary systems". Figure 1 shows some ways of classifying stars, along with associated terminology.
[FIGURE 1: Ways of classifying double stars, with terminology in red and systems of interest shaded.]
One way to determine whether a double star might be binary is to compute how fast the stars are moving relative to each other, and compare that to the escape velocity for the system. In order to know the system escape velocity, we need to know the masses of the stars. Unfortunately, there is no direct way to measure star masses. However, if we know a star's luminosity and we know what type of star it is, we can place it on the HR diagram shown in Figure 2, and this will allow us to estimate its mass by reading it off of the diagram.
[FIGURE 2: HR Diagram. Note the masses shown in purple.]
Analyzing Your Double Star System
First, make your own editable copy of this spreadsheet. The data from the target selection spreadsheet has been copied into the labeled columns. Note that the order of the data columns is different from the order in which you entered the data previously. For identifying stars in catalogs, you want the data for each star on its own line, but to analyze a pair of stars, you need to see the corresponding data values side-by-side (e.g. plx of primary right next to plx of secondary). You will be entering formulas into your copy of the spreadsheet, and using the values for the sample data as a check that you did it correctly. Although you should leave (and include) all the precision in your spreadsheet, please use Format → Number → Custom Number Format to show only 2 decimal places of precision (0.00) in each number and 3 decimal places of precision (0.000) in the corresponding uncertainty ("error").
As you go through the instructions, answer the 4 questions in the "questions" tab of your spreadsheet.
Star Magnitudes and Masses
Column AC: Absolute Gmag of the Primary Star
In column AC, enter a formula for the visual absolute magnitude (Mv) of the star. This is the brightness that the star would appear from a distance of 10pc. The equation relating Mv to the apparent magnitude (m) and parallax (p) of the star is:
Mv = m + 5* (log p + 1)
To enter this formula into the spreadsheet, use the fact that the value you entered for Gaia Gmag in column AA is (approximately) equal to the star's visual magnitude, because Gmag represents the magnitude in the most "green" of the filters on the Gaia spacecraft (actually this is a derived value because Gaia only has blue and red filters on board, but just go with it). The value in column I is its parallax in milliarcseconds (which can be converted to arcseconds by dividing it by 1000). Therefore, in cell AC2, you would type:
=AA2+5*(LOG10(I2/1000) + 1)
The number that shows up in cell AC2 should not change from its initial value of 10.9938808, because that is the absolute Gaia G-magnitude of the star in the sample data. If that happens as expected, then take the little green square at the bottom of cell AC2 and drag it down through all of the other rows of the table to populate those cells with the absolute Gaia G-magnitudes of those stars. For the remaining cells in the spreadsheet, use the same method of checking the result against that given for the sample data. If it matches, copy that formula down to all of the other cells.
Column AD: Absolute Gmag of the secondary star.
Take cell AC2 and copy-paste it into cell AD2, checking that the value in AD2 does not change from its sample data value of 13.62785561.
Columns AE and AF: Estimated star masses.
Columns Y, Z, AC, and AD give you all of the information you need to locate a star in the Gaia HR diagram in Figure 2 below. Look up at the top of the figure to find its spectral class. If your star is on the Main Sequence, estimate its mass within the ranges shown here. Red Giant star masses can range between 0.3 to 8 solar masses; a reasonable guess for a Red Giant would be around 1 solar mass (with high uncertainty!). White dwarf masses are often around 0.6 solar masses.
Go through this exercise for the sample data first to make sure that you are on the right track. Question 1: For the sample data, the secondary, dimmer star of the pair is actually more massive--how can this be?! (Hint: Check out Alex's Astronomy Olympiad screencast from last semester as a cautionary tale about assuming your stars are on the Main Sequence...)
Distances and Separations
Column AG: Distance to primary star
This column will be the distance to the primary star in units of parsecs (pc). Remember that parallax in arcseconds is the inverse of distance in parsecs. If parallax is small, then the stars are very far away from Earth. Note that the parallax values that you entered in columns I and J do not have units of arcseconds--they are listed in milliarcseconds. Do not forget to convert these to arcseconds in your formula. Check your formula against its results for the sample data.
Column AH: Uncertainty in primary distance
This column will be the uncertainty of the distance of the primary. Uncertainty of distance is somewhat tricky since it is computed as an inverse. We will use the tangent line approximation to estimate this. The tangent line approximation says that if y is a function of x, then the error on y is approximately equal to the error on x times the derivative of the function with respect to x. In this case, y = 1/x, so dy/dx = 1/(x^2). Therefore, the error on distance is equal to the error on parallax multiplied by the inverse of the parallax squared. Note that both parallax and error on parallax are given in milliarcseconds, not arcseconds. Therefore, you should incorporate three factors of 1000 into your formula for this column. Check your results against the results for the sample data to make sure you are on the right track.
Columns AI and AJ: Secondary star distance calculations
Columns AI and AJ should repeat the calculations for AG and AH, except that they should do this for the secondary star rather than the primary star.
Column AK: Transverse separation
This column will be the transverse separation of the stars in space. The "transverse separation" is the separation of RA and Dec coordinates at a given distance from Earth, without considering the radial separation of those components. It is the distance analogue of the transverse velocity vector shown in Figure 3.
[FIGURE 3: An illustration of how we split the data associated with a star system into its transverse component (along the "shell" of the celestial sphere corresponding to the distance from Earth) and a radial component (along the line of sight from Earth).]
To find your stars' transverse separation in space, first convert their listed separation from units of arcseconds (symbolized by the double prime symbol, ″) into radians, and then multiply this by the inverse of the listed parallax of your primary star in arcseconds. This gives the separation of the stars' RA and Dec coordinates in units of parsecs, assuming that both stars are at approximately the same radial distance. The full formula is justified using the small angle approximation. It is shown here:
[FORMULA: Transverse separation calculation]
Note that in Google sheets' formulas, pi must be expressed as PI(). Derive a formula using your spreadsheet columns and enter it into column AK, checking your result against the sample data.
Column AL: Radial separation
This column is the radial separation of the stars in space. For example, if one of the stars is 90pc away from Earth and the other is 92pc away from Earth, then their "radial separation" would be 2pc. Derive a formula for this quantity using your spreadsheet columns and enter it into this column, checking your result against the sample data.
Column AM: Three-dimensional separation
This column is the three dimensional separation of the stars in space. To find this, imagine two points in 3-dimensional space. Each point has an x, y, and z coordinate. You can use the distance formula to calculate the 3D separation of the points:
[FORMULA: 3D distance formula with x, y, z coordinates]
But, now suppose that you already know the distance d between the points in the xy plane, and the height h of each in the z direction. Then the formula for the 3D separations of the points is a little simpler:
[FORMULA: Simplified 3D distance formula with d and h]
Can you make an analogy between this and what we did with RA, Dec, and radial separation? Derive a formula for the 3D separation using your spreadsheet columns and enter it into this column, checking your result against the sample data.
Gravitational Analysis
Column AN: Escape velocity
This column is the escape velocity of the system, which is derived in Appendix 1 of this paper and is given by the formula:
[FORMULA: Escape velocity calculation]
Where m1 and m2 are the masses of the primary and secondary stars (columns AE and AF). If the parallax uncertainties do not overlap, then r is the three dimensional separation in space of the stars (column AM). If the parallax uncertainties do overlap, then r is the two dimensional separation of the stars, without the radial component. This is because as noted previously, two stars whose distance uncertainties overlap might be at exactly the same distance from Earth!
Either way, remember to convert the units of solar masses to kg and units of parsecs to meters in order to match them up with the Universal Gravitation constant G! Some conversions are in the spreadsheet if you scroll down. To refer to them in your formula, use the $ sign. For example, $AN$28 is an "absolute reference" to the number in cell AN28, which will not change if you copy-paste the formula that references it into a different cell.
Proper Motion Analysis
Columns AO and AP: Proper motion vector lengths
These columns give the length of the proper motion vectors for the primary and secondary stars, respectively. To find them, take the proper motion in right ascension as x and the proper motion in declination as y, and calculate the total distance as you would normally do for a 2-dimensional vector. Note that for the primary star, the proper motion in RA and Dec is given by columns M and Q. For the secondary star, the proper motion in RA and Dec is given by columns N and R. One useful exercise is to sketch the proper motion vectors on a piece of paper, showing boxes around their tips to represent the errors. Could your stars be moving together through space, or are the directions in which they are moving too different from each other? (Note that proper motion is typically several orders of magnitude greater than orbital motion.)
For example, if the primary star's proper motion in (RA, Dec) is (4 mas/yr ± 0.5mas/yr, 3mas/yr ± 1 mas/yr) and the secondary star's proper motion in (RA, Dec) is (3 mas/yr ± 1 mas/yr, 4 mas/yr ± 0.5mas/yr), then a diagram would look like the one shown in the figure below.
[FIGURE: Proper motion vector diagram showing error boxes]
Within their errors, these stars might be moving together through space as a pair.
Column AQ: Longer proper motion vector
This column gives the length of the longer of the two proper motion vectors in columns AO and AP. So, if the value in column AO is larger, then it should contain the value in column AO. If the value in column AP is larger, then it should contain the value in column AP. Spreadsheet "if statements" have strange syntax; the expression you want is shown below:
=IF(AO2>AP2,AO2,AP2)
Column AR: Relative proper motion vector
This column gives the relative proper motion vector that determines how the stars are moving relative to each other. For this, you want to subtract the vector of the secondary from the vector of the primary and take the length of the resulting vector. Derive a formula for this value in terms of columns M, Q, N, and R and input that into the cell. As always, check your result against the sample data value to make sure that you are on the right track before copy-pasting the formula the rest of the way down the column.
Column AS: Relative proper motion quotient
This column is the quotient of the relative proper motion and the longer of the two proper motion vectors. Calculate this value, check your result, and then think about it.
Question 2: What does the rPM quotient that you calculated in column AS tell you about how the stars in the pair are moving? Why is it important?
Column AT: Proper motion classification
This column gives the classification of your stars based on their proper motions. Because it is a funky spreadsheet "if" statement, the expression is given below. However, although you do not need to formulate the expression yourself, you do need to think about it.
=IF(AS2<0.2,"CPM",IF(AS2>0.6,"DPM","SPM"))
Question 3: Why does the formula for column AT make sense? Note that the meanings of the classifications are given in the table below.
| Classification | rPM range | Meaning |
|---|---|---|
| Common Proper Motion (CPM) | 0 - 0.2 | Stars are moving together through space. |
| Similar Proper Motion (SPM) | 0.2 - 0.6 | Stars are moving through space in a similar way. |
| Different Proper Motion (DPM) | Higher than 0.6 | Stars are not moving together through space. |
Physical Motion Calculation
Column AU: Relative motion in physical units
This column gives the relative motion of the stars, but in physical units rather than angular units. The formula is similar to the one we used in Column AK, except that we need to include a conversion from milliarcseconds to arcseconds and from years to seconds.
[FORMULA: Relative motion calculation with conversion factors]
(where the red values come from Gaia; the black values are conversion factors).
Note well: 1. If we want our physical units to be meters, we must multiply by 1000 (because the expression comes out in km) 2. If parallax is in mas, we must multiply by 1000 (because plx is in the denominator of the expression and is assumed to be arcseconds)
Therefore, you can start with the value in column AR, multiply by 4740 (this collects all of the constants in the conversion) and divide by the parallax of the primary star in column I. (Note that for purposes of simplicity, we are assuming that both stars are at the same distance from Earth in this calculation. The choice of which star's parallax to use here is somewhat arbitrary.)
Column AV: Relative radial motion
In this column, you should calculate the relative radial motion of the stars. Note that you have both of their radial velocities in columns U and V. The units of radial velocity are km/second. If Gaia did not list the radial velocities for the stars, then you should enter 0 in this column.
Column AW: Relative three-dimensional motion
In this column, you should calculate the relative three-dimensional motion of the stars. If you do not have a radial component, then the three-dimensional motion will be the same as the tangential motion. (Elane did something very similar to this in her Astronomy Olympiad problem screencast for OASTR1!)
Column AX: Gravitational bound classification
In this column, you should decide whether the primary and secondary stars in your system are likely to be gravitationally bound based on all of the data in the spreadsheet. Note that if your stars' parallax error bars overlap, they might be exactly the same distance from Earth, which would make the escape velocity much higher. To test this, use column AK instead of AM in the denominator of the escape velocity function!
Question 4: Give a brief (2-sentence) description of which values you compared to arrive at your classifications in column AX, and why you used those values.
Next Steps
After completing this analysis, you'll have determined whether your double star system is likely to be physically related and possibly gravitationally bound. This information will be crucial for interpreting the historical measurements and your own observations, and will form a key part of your scientific paper.