SOME MATHS FOR HOME ASTRONOMERS
-----------------------------
John Pazmino
NYSkies Astronomy Inc
nyskies@nyskies.org
www.nyskies.org
2020 June 15
Introduction
----------
Much of the avoidance of maths by home astronomers are the smbols
and notations in formulae. Unlike in old times, we can not skip over
the maths and live off of only descriptions and pictures in our
profession.
We now easily obtain and peruse the techincal litteraure of
astronomy directly from the campus astronomers, thanks to heir
publication in the web. We have devices, like calculettes, phone
applications, computer software, that do the numberical calculations
in the league of campus astronomers. We interact with many other
sciences newly integrated into astronomy, with their own technical
procedures. We avail of prime or processed material from astrophysical
and eophysical spaceprobes.
These, and likely other, factors, coming into home astronomy
require us to have an acquaintance and awareness of the mathematics,
even if we do not manipulate them or develope original owrks.
Here I give some guidance for many aspects of maths that you may
have bypassed readers, clarifying their features. The slate is not at
al complete and some details, by their complexity, are only sketched
out here.
Symbols + and -
-----------
By history we use the same symbols +, - for the signum the
add/subtract operation s. The two meanings are utterly separate and
must never be mixed up.
The + and - signa are an integral part of their number and can
never be omitted or ignored. When filling a formula with a number its
signum should be explicitly written. If need be, the number and its
signum can be roped in parens to keep them distinct from the + and -
operations.
This procedure can lead to the apparent duplication of a symbol,
like
c = +a - (-b)
The - in the clear is the subtract symbol. It instructs to subtract b
from a. The - in parens belongs to number b as its signum. b is a
negative number. It is inside the parens to make sure it stays with b
and does not creep into the operational symbols of the formula.
The operational - is part of the formula, regardless of the signa
of a and b.
A positive number is commonly written without its + signum. This
is risky. It's safer to write in the signum.
Symbol *
------
From long experience with computer languages, the char x or X for
'multiply' is displaced by *. It was far too easily assimilated as an
alphabetic of a number's name. Also displaced is the abutting of two
numbers for multiplication, which looks like a name for a new number.
The * is now generally used in text outside of computer coding,
even hand text. ,
c = a * b, (in the stead of axb and ab)
This instructs to multiply number a by number b.
Symbol /
------
The divide operation is indicated by both the / and the
underline____. The ___ is drawn long enough to span the char length of
dividend and divisor. In computer code only the / is available because
all chars are written linearly.
All kinds of trouble come from misuse of the divide symbol, partly
from the careless writing of the operation and loose maths skill.
Ideally a formula can be written to include only one divide
operation. All terms on the left are evaluated, then all those on the
right. Finally the two sides go into the divide. In many cases this
results in an erroneous formula.
The way to keep trouble away is thru liberal use of parens to
force the sequence of operations around the divides. Or, with parens,
separate the several divides into their own 'cells'.
A notorious situation of misplaced divides is when the formula has
piggyback divides. The impulse is
c = a / b / d
This can mean
c = (a / b) / d or a / (b / d)
Here the parens force the sequence of the divides. A better way to
avoid trouble is to compose the formula to have as few divides as
practical. We could have
c = a / (c * d) or c = (a * d) / c
A sanity check can be done by inserting simple numbers into the
formula and working it thru the possible variations of the divides.
One version yields the correct sequence while the others hand up
ridiculous ones.
Symbol ^
------
This indicates raising a number to a power. In hand work the power
is a small superscript after the number. Since virtually all wrodprocs
make it clumsy to insert superscripts and computer code don't have
superscripts, the number and its power are separated by ^.
c = a ^ b
number a is raised by the power b.
A rare alternative is the **, c = a ** b. This is just about never
used in hand work.
Whem the ppwer is a fraction. it is a root
a ^ (1 / b) = b-root(a)
tells you to take the b-th root of a. The 2nd root, the square root,
is commonly shown as
c = sqrt(a) (with some variation in the abbbreve)
A negative power means the resulting number is then divided into
one, or is then a reciprocal.
a ^ (-b) = 1 / (a ^ b )
Powers and roots, specially nonintegral ones, are handled thru
calculetttes and other maths devices. Don't work them out by hand.
Sequence of Operation
-------------------
In the ideal case operations are carried out in the order laid out
by parens. It is common to have formulae with few or no parens and you
must figure out the sequence of operations. Doing them out of their
intended order yields erroneous results.
The fall-back procedure does the higher level operations first,
then working to the lower level ones. The order is power/root,
multiply/divide, add/subtract.
c = a * b + f / d ^ e
is processed first by d ^ e, then f / (d ^ e), a * b, (a * b) + ((f /
d ^ e). To force the proper sequence, put in parens, like
c = (a * b) + (f / (d ^ e))
c = (a * (b + (e / d) ^ e))
The two are NOT equivalent. The parens make the operations procede in
different sequence..
Parens
-
Short for 'parentheses', parens enclose operations that are
performed within them before those outside them. The general rule is
to do the innermost parens first, then work thru the outer ones.
Insert parens to make sure the operations in the formula are done
in the correct order.
Parens are specially important to distinguish between + and - as
signa and operations. An other important case is showing an operation
done on the argument of a function and on the very function.
Parens must be balanced, equal count of ( and ) in a formula.
Verify this by stepping thru the formula and tallying each ( and ).
Unbalanced parens can led to uncertain operations in the formula.
c = sin(a) ^ 3 or sin(a ^ 3)
c = log(a) ^ 2 or log(a ^ 2)
Some formulae put the power against the name of the function, like
sin^2(a). This is acceptable but you must recognize that the power is
done on the value of the function.
Powers of Ten
-----------
Large and small numbers are commonly written as a base number
multiplied by a tens power.
72,000,000 -> 72X10^6.
10^6 is one million and this is multiplied by 72. This is for typed
text but clumsy and often hard to follow in hand text.
Banking off of computer code, where the X and 6 in this example can
not be properly handled, the e-form is now prevalent. It is easy to
write by hand.
The base and its tens power are separated by 'e' or 'E'.
72X10^6 -> 72e6
he ;e' bodily suplants the 'X10^'. The number is spoken as '72 e 6',
simpler than the older '72 times 10 to the 6th power'.
All the math rules for powers apply to the e-form. It is specially
important to mind the signum of the power. In critical cases th signum
is deliberately written.
You may shift the power by shifting the decimal point of the base
number in the opposite sense. A step higher in power is a decimal
shift to the left; lower, right. Missing places in the decimal shift
are filled with zeros.
72e6 = 7200e4 = 0.0072e10
Inequalities
-----------
An inequality compares tow numbers that are not actually equal,
telling which is the greater. This function helps limit the range of
possible results of calculations. The common inequalities are
<, 'less than', arrow points from larger to the smaller number
>, 'greater than'
>=, 'equal or greater then', also =>, >_ (> underlined >)
<=, 'equal o less than', also =<, <_ (< underlined <)
<>, 'not equal to', also =!, =/ (equal with a slash thru it)
>>, 'much greater than', by an order or more
<<, 'much less tan', by an order or more
The comparison is the left number against the right number. Note
that
a > b <-> b < a
a <= b <-> b => a
and similarly for the others.
The behavior of inequalities can be tricky when they shift around
in a formula . It's wise to mentally insert simple integers into the
numbers and see how the relations change as you manipulate the
formula. In general move BOTH numbers with the inequality operator as
a unit.
Logarithms
--------
A logarithm of a given number is the power of a base number that
makes the given number. For the base 10, log(23) = 1.3617 because
raising 10 to the 1.3617 power yields 23.
There are two schemes of logs, base 10 and base epsilon or e,
which is 2.71828. This number is a number describing processes and
activity in nature. Logs on base 10 are the Briggs or common scheme;
w, Napier or natural logs. Both are found in astronomy.
a common log is denoted by log(a) or log109a). A natural log is
ln(a) altho in some formula the formula's text may advise that the its
'log)a)' is the natural logarithm.
Likely the logs you come across are in formulae laid out to accept
input in log form. Do the maths on the logs as indicated within the
formula.
Finding the logarithm of a number or the number for a given
logarithm by maths devices. The need for tables of logarithms is just
about gone. Older astronomers keep the tables they had from school but
probably will never go and get a new one.
A caution is the notation for the antilog, instructing to get the
number for the given log. The usual maths symbol is
b = log^-1(a)
This does NOT mean 1 / log(a)! Reciprocal log is written
(log(a))^-1 or 1 / ln(a)
Much better notations are alog, alg, aln (natural log).You may also
see
aalog(a) = 10 ^ a
aln(a) = epsilon ^ a
to get the regular number a from its log or ln b. This uses the raw
definition of a logarithm.
For the occasion you want to convert between natural and common
logs
log(a) = ln(a) / ln(10O)
= ln(a) / 2.3026
ln(a) = log(a) / log(2.71828)
= log(a) / 0.43429
Factorials
--------
A factorial is a number composed of the multiply of all the
positive integers from 1 to the number. It is denoted by A! or
fact(A). The notation is occasionally spoken as 'a bang' after the
name of the ! symbol.
A! = 1 * 2 * 3 * 4 * ...* A
Fact(8) is 40,320. Factorials grow humongous very rapidly. 10! is,
about3.6 million and 15! is about 1.3 trillion. A maths device may cap
the highest number allowed for taking its factorial.
Fact(1) and fact(0) are for technical reasons set to 1.
There is factorial of a negative number. To state a negative
factorial
b = -fact(a) or -(a!)
Factorials come up in formulae dealing with statistics ,
probabilities, combinations & permutations.
Vectors
-----
A vector is a number that has, in addition to other properties, a
direction, alignment, orientation. The wind speed is a vector because
besides the size, like meter/second or kilometer/hour, it has
direction, like from northeast. (winds are stated with the direction
thry blow FROM, not to.)
The name of a vector is in typed text a bold char, char topped by
a small arrow, a fancy style of char. In linear text 'vec' before the
name demotes a vector.
A vector's parameters may be either its components along the c-
and y- coordinate axes or its full length and angle against the x-
axis. (We here examine only two-dimensional vectors.)
vecA(ax, ay) <-> vecA(ar, ap)
where ax, ay are the lengths along the x- and y-axes; ar, full length;
ap angle against the x-axis. Which method to use depends on the
instant situation. At times the method is already set in the formula.
You may convert between forms
r = sqrt(x^2 + y^2), p = atan(y/x) = 90deg - atan(x / y)
In general, use the angle method with the smaller tangent.
x = r * cos(p), y = r * sin(p)
Adding vectors is easiest done with the x,y parameters.
vecA(ax, ay) + vecB(bx, by) = vecC(ax + bx, ay + by)
= vecC(cx, cy)
from which the r,p form can be calculated.
To subtract vectors, flip the signa of the second vector's x,y and
so an add.
vecA(ax, ay) - vecB(bx, by) = vecA(ax, ay) + vecB(-bx, -by)
There are two ways to multiply vectors, the dot and cross
methods. The actual process can't be properly detailed here but some
features of the two methods must be appreciated.
The dot multiply is in typed work denoted by a fat dot between the
vectors. In linear work we write 'dot'.
The dot, or scalar, multiply yields a new regular number, with no
vector properties. The order of the multiply doesn't matter, either
gives the same result.
vecA dot vecB = vecB dot vecA -> C (regular number)
The method uses the angle between the two vectors, not the angles
they stand against the x-axis. The maximum value of the dot multiply
occurs when the vectors are lined up with angle 0 between them. The
minimum value, which is zero, occurs for angle of 90 degrees.
The other method is the cross or vector multiply, written as a fat
or large 'X between the vectors in typed text. 'Cross' is used in
linear work.
The cross multiply also uss the angle between the vectors but the
answer is a new vector, not a regular number. This new vector is
perpendicular to the plane of the original vectors. The order of
multiply counts. Swopping the vectors makes the resultant vector point
in opposite direction.
vecA cross vecB -> vecC (per(pendicuarto vecA and vecB)
vecB cross vecA -> -VecC (opposite in direction from vecC)
The cross multiply has its maximum value when the angle between
the vectors is 90 deg. It zeros out when the angle is 0 deg.
The direction of vecC is given by the right-hand-rule. Mentally
curl the right hand -- not the left! -- around the two vectors as if
to push vecA into vecB, closing the intervening angle. The thumb
points in the direction of vecC.
If you curl the hand to push vcB into vecA, vecC points in the
opposite direction. This ts the effect of swopping the order of the
multiply.
There is no choosing between the dot and cross methods. They apply
in wholly different situations. You must recognize the one applicable
to the instant case.
Angles
----
We use two schemes of angle measure. The common one is the degree-
minute-second, d-m-s, almost exclusively stated for astronomy angles.
The other is the radian measure, where the circumference of a
circle is paced off in arcs equal to the circle's radius. Since the
circumference is 2*pi radii in length, there are 2*pi arcs. The six
large arcs span an angle of one radian, with a small leftover piece.
Phrased in other words, the circle contains 2*pi radians of angle,
like it contains 360 degrees of angle.
One radian is abut 57.2958 degree or 205,265 second. For rough
work one radian is 200,000 seconds.
radians = degree / (57.2958 deg/rad)
= seconds / (205265 sec/rad)
degrees = radians * (57.2958 deg/rad)
seconds = radians * (205265 sec/rad)
For maths the d-m-s angle is way to clumsy to work with. Convert
it to degree & decimal, d-d.
d-d = degrees + (minutes / 60) + (seconds / 3600)
it's your choice to convert the answer in d-d back to d-m-s
d-m-s = degree, decimal * 60), (leftover dec * 3600)
First set aside the integer part of the d-d as the whole degrees.
Then multiply the decimal part by 60 for the minutes. The integer
part of this is the whole minutes.
The leftover decimal is multiplied by 3600 for the seconds. Any
further decimal is the fraction of seconds.
Read carefully a formula with angles. You must use the correct
form, d-m=sm, s-s, or radians.
In astronomy angles can wrap around the circle beyond 360 degrees
in either sense, + or -. You may have to toss whole revolutions from
the final answer.
Trig Functions
------------
Finding the trig function of a given angle is handled by the
calculette. An angle has a single unique value for its trig functions.
The inverse is not so simple. A given trig number has TWO angles
and the calculette returns ONE of them. YOu must study the operating
manual to learn which angle is returned for each of the functions.
You must separately figure which is the correct angle for the
instant situation and shift the returned angle to it. Make a sketch of
the problem and examine where the angles lie.
This chart helps follow the behavior of trig functions around the
360 degrees
---------------------------------------------
| 0 |+/- | 90 |+/-| 180 |+/-| 270 |+/-| 360
--------+----+----+----+---+-----+---+-----+---+------
cosine | +1 | + | 0 | - | -1 | - | 0 | + | +1
sine | 0 | + | +1 | + | 0 | - | -1 | - | 0
tangent | 0 | + |+inf | | | |+inf | + | 0
|-inf | - | 0 | - |-inf |
----------------------------------------------------
The +/-/ columns show the signum of the function between adjacent
quadrant angles. Note carefully that the tangent flips signum when it
crosses 90 and 270 degrees.
The formal maths notation for getting the angle for the given
trig function is, for tangent as example,
a = tan^--1(b)
This is NOT the reciprocal of the tangent. An intended reciprocal is
a = (tan(b))^-1 or 1 / tan(b)
Far better notations are arctan, atan, atn, and parallel ones for
the sine and cosine.
There are six main functions, but three are reciprocals of the
others, making them rarely used for themselfs. Calcuettes generally
carry only the prime ones
secant(a) = 1 / cosine(a)
cosecant(a) = 1 / sine(a)
cotangent(a) = 1 / tangent(a)
Derivatives and Integrals
-----------------------
There really is no simple way to work with integrals and
derivatives without some dedicated tuition in calculus.
The nest to do here is recognize these features in maths and rely
on circumstant text to explain them.
A derivative is the rate of change of A during a change in B. A
must be a function of B, either explicitly in the formula or in
associated text. A derivative is recognized by notations like
d(A) / d(B), delta(A) / delta(B) (the Greek char), A', A. (dot on
top of A)
` The last two do not show B directly but there must be text twlling
what it is.
The first is easiest to read as symbolicly saying 'change in A for
a unit chage in B'.
Derivatives may be nested to second or more levels
d2(A) / d(B)2, d3(A) / (dB)3
These refer to a derivative of a drivative, third level derivative,
and many deeper ones.
The ' and . notations are not commonly extended to higher leels of
derivative. The d or delta forms are use as illustrated above.
An integral is a summation of A over a range of B. A must be a
function of B in either the formula or associated text.
C = S(A) * dB
The 'S' is drawn as a tall lazy 'S' before A. In some cases the
summation is shown by a tall capital sigma. The d*B( is then usually
written as delta*B)
c = SIGMA(A) * d(B), SIGMA(A) * delta(B)
In some cases the range of B is shown as the upper and lower
values at the tip and bottom of the S or SIGMA
y y
S(A) * d(B), SIGMA(A) * d(B) ,
x x
The instruct to add the value of A thru the range of B from x to y.
Integrals may be nested to show a summation ot a summation, to
several levels.
C = SSS(B) * dD * dE *dF
The inner integral is evaluated first, then the otherstowaes the
outermost one. A range on an integral symbol applies only to that
integral, not to any of the others.
For home astronomy there are probably few instances to somehow
figure out how to evaluate these features in a formula. The typical
case is the formula is derived or composed and then the final
iteration is presented.
Units of Measure
--------------
Formulae often relate numbers with units of measure. The usual
procedure is to do the maths on just the numbers and affix the
expected final units to the answer. One strong reason for this
procedure is that few calculkttes have yumbolic operations. They work
only on numbers.
Ignoring the units in a formula, by any excuse, is dangerous!
It is poorly appreciated that maths apply to the units along with
the numerical value. To keep the units and number together as they
procede thru the formula, encase both in parens.
Maths operating on units yield a symbolic representation of the
opeation. A couple simplified notationsa re widely used. Within the
units a multiply is a dot . and the ^ for power/root is omitted.
Divides are routinely shown only by the /.
c = (a N/kg) * (b m)
= (a * b) N.m/kg
e = (m kg) * (c m/s) ^ 2
= (m * c ^ 2) kg.m2/s2 (which equals (m * c ^ 2) J)
In complex formulae some symbols look like units and can cause
major errors if mixed up. One trick is to use bumpers [ ] to encase
units.
Doing the maths on just the units helps rap potential mistakes.
TIf the units don't come out right, you did something wrong. The
converse isn't always true because you still could make a maths
mistake with the numbers.
This situation comes often in converting among units. The general
conversion instruction is 'to get unit A multuply unit B by number C'
or '... divide by number C'. You will forget which to do for a given
instance.
To convert 150 inch to meter so we multiply ?, or divide?, by by
39.39? Try multiply
A meter = (B inch) * 39.37
= (150 inch) * 39.37
= 5,905.5 meter
It is glaringly wrong! 150 inch is about the length of a msall car
while 5,905.5 meter is the width of a town. The two are NOT equal.
The right way to instruct the conversion is 'to change inch to meter,
divide by 39.37 inch/meter'. The correct formula is
A meter = (B inch) / (39.37 inch/meter)
= (150 inch) / (39.37 inch/meter)
= 3.81 inch.meter/inch
= 3.81 meter.
Doing the maths on BOTH the number and the units self-checks the
operation. The weong answer would be 5,905.5 inch2/meter, immediately
a nonsense result.
Systems of Units
--------------
We use in astronomy and other sciences two major systems of
units. The current formal one is the International System, SI (from
the french phrase), whose base units include the meter, kilogram,
second. The other is the older, still widely used, CGS, standing for
its base units of centimeter, gram, second.
In these two schemes, units of the same kind are ten-folds of each
other, They usually can be translated at sight. Hopping between
systems is vital for formulae composed to accept units in a one while
you have numbers with units in the other.
Some sciences work with non-standard units, which may or not derive
from the metric system. Astronomy has several such units, like the
angstrom, parsec, jansky, solar mass.
In both SI and CGS many combinations of unit are given an honorary
name like newton for kg.m/s2. According as the situation to hand, you
may swo out the honorary name for its component units and vice versa.
There seems to be no formal procedure to choose combinations for
an honorary name. SI and CGS have their own sets of honorary names.
Multiples and Prefixes
--------------------
The metric system makes heavy use of prefixes for multiples, and
submultiples, of the their units. We may say 'ten kilometers' or
'10,000 meters' because 'kilo-' means '1.l000'.
As convenience calls a prefix amy swop for a multiple, and vice
versa. This comes in handy if numbers span a wide range of a unit and
it's simpler to do maths on one form of the unit.
The prefix enters the maths with the unit. Do not apply the maths
to each separately.
When shifting from multiple to prefix, mind well the movement of
the decimal point in the number. It may slide beyond the written chars
of the number. The missing places for the shift are filled with zeros.
17.2 litter = 17,200 cm3 (1 liter is 1,000 cm3; two zeros fill
the missing places)
86 g = 0.086 kg (the missing place is filled with one zero)
It can be handy to piggyback predixes, altho this is not a feature
in the metric system. You may come onto 'millimicrogram' during a
calculation. You may at the end of the work collapse the piggybackers
into a single prefix, here 'nanogram'.
Conclusion
--------some of the maths features i discuss here are brief, with
bypassed procedures and details. It really pays to get a maths review
book for 10th and 11th grade mathematics. This covers plane geometry,
algebra, trigonometry, statistics, logarithms,coordinate analysis .
High school maths generally do not include solid geometry or calculus,
altho enhanced course may do so.
Many review books have samples from state or city qualifying
tests. work thru the examples. From the web online tuitions or actual
school courses walk you thru the maths. Search for for 'plane
geometry' and visit sites run by schools. Try those in remedial
college courses